Theorem pprodcnveq 35901

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 35900	. 2 ⊢ pprod(𝑅, 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2		dfpprod2 35900	. . . 4 ⊢ pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
3	2	cnveqi 5854	. . 3 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ◡((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
4		cnvin 6133	. . 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 35776	. . . . 5 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6		cnvco1 35776	. . . . . 6 ⊢ ◡(◡𝑅 ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ 𝑅)
7	6	coeq1i 5839	. . . . 5 ⊢ (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8		coass 6254	. . . . 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 35776	. . . . 5 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11		cnvco1 35776	. . . . . 6 ⊢ ◡(◡𝑆 ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ 𝑆)
12	11	coeq1i 5839	. . . . 5 ⊢ (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13		coass 6254	. . . . 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 4193	. . 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 1540  Vcvv 3459   ∩ cin 3925   × cxp 5652  ^◡ccnv 5653   ↾ cres 5656   ∘ ccom 5658  1^st c1st 7986  2^nd c2nd 7987  pprodcpprod 35849

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-txp 35872  df-pprod 35873

This theorem is referenced by:  brpprod3b  35905