Theorem pprodcnveq 35871

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 35870	. 2 ⊢ pprod(𝑅, 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2		dfpprod2 35870	. . . 4 ⊢ pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
3	2	cnveqi 5838	. . 3 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ◡((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
4		cnvin 6117	. . 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 35746	. . . . 5 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6		cnvco1 35746	. . . . . 6 ⊢ ◡(◡𝑅 ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ 𝑅)
7	6	coeq1i 5823	. . . . 5 ⊢ (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8		coass 6238	. . . . 5 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
9	5, 7, 8	3eqtri 2756	. . . 4 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10		cnvco1 35746	. . . . 5 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11		cnvco1 35746	. . . . . 6 ⊢ ◡(◡𝑆 ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ 𝑆)
12	11	coeq1i 5823	. . . . 5 ⊢ (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13		coass 6238	. . . . 5 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
14	10, 12, 13	3eqtri 2756	. . . 4 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
15	9, 14	ineq12i 4181	. . 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 2756	. 2 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
17	1, 16	eqtr4i 2755	1 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)

Syntax hints:   = wceq 1540  Vcvv 3447   ∩ cin 3913   × cxp 5636  ^◡ccnv 5637   ↾ cres 5640   ∘ ccom 5642  1^st c1st 7966  2^nd c2nd 7967  pprodcpprod 35819

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-txp 35842  df-pprod 35843

This theorem is referenced by:  brpprod3b  35875