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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 35856 | . 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) | |
| 2 | df-txp 35855 | . 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 2765 | 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 1540 Vcvv 3480 ∩ cin 3950 × cxp 5683 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 1st c1st 8012 2nd c2nd 8013 ⊗ ctxp 35831 pprodcpprod 35832 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-txp 35855 df-pprod 35856 |
| This theorem is referenced by: pprodcnveq 35884 |
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