Theorem pprodcnveq 36054

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 36053	. 2 ⊢ pprod(𝑅, 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2		dfpprod2 36053	. . . 4 ⊢ pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
3	2	cnveqi 5822	. . 3 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ◡((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
4		cnvin 6101	. . 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 35932	. . . . 5 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6		cnvco1 35932	. . . . . 6 ⊢ ◡(◡𝑅 ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ 𝑅)
7	6	coeq1i 5807	. . . . 5 ⊢ (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8		coass 6223	. . . . 5 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
9	5, 7, 8	3eqtri 2762	. . . 4 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10		cnvco1 35932	. . . . 5 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11		cnvco1 35932	. . . . . 6 ⊢ ◡(◡𝑆 ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ 𝑆)
12	11	coeq1i 5807	. . . . 5 ⊢ (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13		coass 6223	. . . . 5 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
14	10, 12, 13	3eqtri 2762	. . . 4 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
15	9, 14	ineq12i 4169	. . 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 2762	. 2 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
17	1, 16	eqtr4i 2761	1 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)

Syntax hints:   = wceq 1542  Vcvv 3439   ∩ cin 3899   × cxp 5621  ^◡ccnv 5622   ↾ cres 5625   ∘ ccom 5627  1^st c1st 7931  2^nd c2nd 7932  pprodcpprod 36002

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-txp 36025  df-pprod 36026

This theorem is referenced by:  brpprod3b  36058