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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 35836 | . 2 ⊢ pprod(𝐴, 𝐵) = ((𝐴 ∘ (1st ↾ (V × V))) ⊗ (𝐵 ∘ (2nd ↾ (V × V)))) | |
2 | df-txp 35835 | . 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 2762 | 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 1536 Vcvv 3477 ∩ cin 3961 × cxp 5686 ◡ccnv 5687 ↾ cres 5690 ∘ ccom 5692 1st c1st 8010 2nd c2nd 8011 ⊗ ctxp 35811 pprodcpprod 35812 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1776 df-cleq 2726 df-txp 35835 df-pprod 35836 |
This theorem is referenced by: pprodcnveq 35864 |
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