In summary: The interplay of meaning and axiomatic machine mathematics, captured by the difference between $\models$ and $\vdash$, is a subtle and interesting thing.
What is the meaning of the $c$ in $C_c^ {\infty} (\mathbb {R})$?
In context that looks the same as =: (meaning the left and right sides are equal and the left side is the definition of the right side).
sequences and series - What is the meaning of “=::” in mathematics ...
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