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Theorem pprodcnveq 35865
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 35864 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2 dfpprod2 35864 . . . 4 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
32cnveqi 5888 . . 3 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
4 cnvin 6167 . . 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 35739 . . . . 5 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6 cnvco1 35739 . . . . . 6 (𝑅 ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ 𝑅)
76coeq1i 5873 . . . . 5 ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8 coass 6287 . . . . 5 (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
95, 7, 83eqtri 2767 . . . 4 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10 cnvco1 35739 . . . . 5 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11 cnvco1 35739 . . . . . 6 (𝑆 ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ 𝑆)
1211coeq1i 5873 . . . . 5 ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13 coass 6287 . . . . 5 (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
1410, 12, 133eqtri 2767 . . . 4 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
159, 14ineq12i 4226 . . 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 2767 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
171, 16eqtr4i 2766 1 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3478  cin 3962   × cxp 5687  ccnv 5688  cres 5691  ccom 5693  1st c1st 8011  2nd c2nd 8012  pprodcpprod 35813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-txp 35836  df-pprod 35837
This theorem is referenced by:  brpprod3b  35869
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