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Theorem pprodcnveq 32503
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 32502 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2 dfpprod2 32502 . . . 4 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
32cnveqi 5500 . . 3 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
4 cnvin 5757 . . 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 32163 . . . . 5 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6 cnvco1 32163 . . . . . 6 (𝑅 ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ 𝑅)
76coeq1i 5485 . . . . 5 ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8 coass 5873 . . . . 5 (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
95, 7, 83eqtri 2825 . . . 4 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10 cnvco1 32163 . . . . 5 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11 cnvco1 32163 . . . . . 6 (𝑆 ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ 𝑆)
1211coeq1i 5485 . . . . 5 ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13 coass 5873 . . . . 5 (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
1410, 12, 133eqtri 2825 . . . 4 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
159, 14ineq12i 4010 . . 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 2825 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
171, 16eqtr4i 2824 1 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  Vcvv 3385  cin 3768   × cxp 5310  ccnv 5311  cres 5314  ccom 5316  1st c1st 7399  2nd c2nd 7400  pprodcpprod 32451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-txp 32474  df-pprod 32475
This theorem is referenced by:  brpprod3b  32507
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