Theorem pprodcnveq 36860

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 36859	. 2 ⊢ pprod(𝑅, 𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2		dfpprod2 36859	. . . 4 ⊢ pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
3	2	cnveqi 5864	. . 3 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ◡((◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))))
4		cnvin 6153	. . 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 36739	. . . . 5 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6		cnvco1 36739	. . . . . 6 ⊢ ◡(◡𝑅 ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ 𝑅)
7	6	coeq1i 5848	. . . . 5 ⊢ (◡(◡𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8		coass 6290	. . . . 5 ⊢ ((◡(1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
9	5, 7, 8	3eqtri 2790	. . . 4 ⊢ ◡(◡(1st ↾ (V × V)) ∘ (◡𝑅 ∘ (1st ↾ (V × V)))) = (◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10		cnvco1 36739	. . . . 5 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11		cnvco1 36739	. . . . . 6 ⊢ ◡(◡𝑆 ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ 𝑆)
12	11	coeq1i 5848	. . . . 5 ⊢ (◡(◡𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13		coass 6290	. . . . 5 ⊢ ((◡(2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
14	10, 12, 13	3eqtri 2790	. . . 4 ⊢ ◡(◡(2nd ↾ (V × V)) ∘ (◡𝑆 ∘ (2nd ↾ (V × V)))) = (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
15	9, 14	ineq12i 4118	. . 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 2790	. 2 ⊢ ◡pprod(◡𝑅, ◡𝑆) = ((◡(1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ (◡(2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
17	1, 16	eqtr4i 2789	1 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∩ cin 4092   × cxp 5656  ^◡ccnv 5657   ↾ cres 5660   ∘ ccom 5662  1^st c1st 8033  2^nd c2nd 8034  pprodcpprod 36808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-txp 36831  df-pprod 36832

This theorem is used by:  brpprod3b  36864