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Theorem pprodcnveq 35159
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
StepHypRef Expression
1 dfpprod2 35158 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2 dfpprod2 35158 . . . 4 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
32cnveqi 5873 . . 3 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
4 cnvin 6143 . . 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 35033 . . . . 5 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6 cnvco1 35033 . . . . . 6 (𝑅 ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ 𝑅)
76coeq1i 5858 . . . . 5 ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8 coass 6263 . . . . 5 (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
95, 7, 83eqtri 2762 . . . 4 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10 cnvco1 35033 . . . . 5 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11 cnvco1 35033 . . . . . 6 (𝑆 ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ 𝑆)
1211coeq1i 5858 . . . . 5 ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13 coass 6263 . . . . 5 (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
1410, 12, 133eqtri 2762 . . . 4 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
159, 14ineq12i 4209 . . 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)))))
163, 4, 153eqtri 2762 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
171, 16eqtr4i 2761 1 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  Vcvv 3472  cin 3946   × cxp 5673  ccnv 5674  cres 5677  ccom 5679  1st c1st 7975  2nd c2nd 7976  pprodcpprod 35107
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-11 2152  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-txp 35130  df-pprod 35131
This theorem is referenced by:  brpprod3b  35163
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