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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 34827 | . 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) | |
2 | df-txp 34826 | . 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 2761 | 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 1542 Vcvv 3475 ∩ cin 3948 × cxp 5675 ◡ccnv 5676 ↾ cres 5679 ∘ ccom 5681 1st c1st 7973 2nd c2nd 7974 ⊗ ctxp 34802 pprodcpprod 34803 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-txp 34826 df-pprod 34827 |
This theorem is referenced by: pprodcnveq 34855 |
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