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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfpprod2 | Structured version Visualization version GIF version |
Description: Expanded definition of parallel product. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
dfpprod2 | ⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pprod 34896 | . 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) | |
2 | df-txp 34895 | . 2 ⊢ ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V))))) | |
3 | 1, 2 | eqtri 2760 | 1 ⊢ pprod(𝐴, 𝐵) = ((◡(1st ↾ (V × V)) ∘ (𝐴 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝐵 ∘ (2nd ↾ (V × V))))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 Vcvv 3474 ∩ cin 3947 × cxp 5674 ◡ccnv 5675 ↾ cres 5678 ∘ ccom 5680 1st c1st 7975 2nd c2nd 7976 ⊗ ctxp 34871 pprodcpprod 34872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-cleq 2724 df-txp 34895 df-pprod 34896 |
This theorem is referenced by: pprodcnveq 34924 |
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