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Theorem pprodcnveq 35884
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 35883 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2 dfpprod2 35883 . . . 4 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
32cnveqi 5885 . . 3 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
4 cnvin 6164 . . 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 35759 . . . . 5 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6 cnvco1 35759 . . . . . 6 (𝑅 ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ 𝑅)
76coeq1i 5870 . . . . 5 ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8 coass 6285 . . . . 5 (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
95, 7, 83eqtri 2769 . . . 4 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10 cnvco1 35759 . . . . 5 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11 cnvco1 35759 . . . . . 6 (𝑆 ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ 𝑆)
1211coeq1i 5870 . . . . 5 ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13 coass 6285 . . . . 5 (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
1410, 12, 133eqtri 2769 . . . 4 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
159, 14ineq12i 4218 . . 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 2769 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
171, 16eqtr4i 2768 1 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  Vcvv 3480  cin 3950   × cxp 5683  ccnv 5684  cres 5687  ccom 5689  1st c1st 8012  2nd c2nd 8013  pprodcpprod 35832
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-txp 35855  df-pprod 35856
This theorem is referenced by:  brpprod3b  35888
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