Theorem pprodcnveq 35884

Description: A converse law for parallel product. (Contributed by Scott Fenton, 3-May-2014.)

Assertion

Ref	Expression
pprodcnveq	⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)

Proof of Theorem pprodcnveq

Step	Hyp	Ref	Expression
1		dfpprod2 35883	. 2 ⊢ pprod(𝑅, 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2		dfpprod2 35883	. . . 4 ⊢ pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
3	2	cnveqi 5885	. . 3 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ◡((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
4		cnvin 6164	. . 3 ⊢ ◡((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V))))) = (◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
5		cnvco1 35759	. . . . 5 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6		cnvco1 35759	. . . . . 6 ⊢ ◡(◡𝑅 ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ 𝑅)
7	6	coeq1i 5870	. . . . 5 ⊢ (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8		coass 6285	. . . . 5 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
9	5, 7, 8	3eqtri 2769	. . . 4 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10		cnvco1 35759	. . . . 5 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11		cnvco1 35759	. . . . . 6 ⊢ ◡(◡𝑆 ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ 𝑆)
12	11	coeq1i 5870	. . . . 5 ⊢ (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13		coass 6285	. . . . 5 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
14	10, 12, 13	3eqtri 2769	. . . 4 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
15	9, 14	ineq12i 4218	. . 3 ⊢ (◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V))))) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
16	3, 4, 15	3eqtri 2769	. 2 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
17	1, 16	eqtr4i 2768	1 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)

Syntax hints:   = wceq 1540  Vcvv 3480   ∩ cin 3950   × cxp 5683  ^◡ccnv 5684   ↾ cres 5687   ∘ ccom 5689  1^st c1st 8012  2^nd c2nd 8013  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-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-txp 35855  df-pprod 35856

This theorem is referenced by:  brpprod3b  35888