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Theorem pprodcnveq 33344
 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 33343 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
2 dfpprod2 33343 . . . 4 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
32cnveqi 5744 . . 3 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
4 cnvin 6002 . . 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 32995 . . . . 5 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V)))
6 cnvco1 32995 . . . . . 6 (𝑅 ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ 𝑅)
76coeq1i 5729 . . . . 5 ((𝑅 ∘ (1st ↾ (V × V))) ∘ (1st ↾ (V × V))) = (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V)))
8 coass 6117 . . . . 5 (((1st ↾ (V × V)) ∘ 𝑅) ∘ (1st ↾ (V × V))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
95, 7, 83eqtri 2848 . . . 4 ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) = ((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V))))
10 cnvco1 32995 . . . . 5 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V)))
11 cnvco1 32995 . . . . . 6 (𝑆 ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ 𝑆)
1211coeq1i 5729 . . . . 5 ((𝑆 ∘ (2nd ↾ (V × V))) ∘ (2nd ↾ (V × V))) = (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V)))
13 coass 6117 . . . . 5 (((2nd ↾ (V × V)) ∘ 𝑆) ∘ (2nd ↾ (V × V))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
1410, 12, 133eqtri 2848 . . . 4 ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))) = ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V))))
159, 14ineq12i 4186 . . 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 2848 . 2 pprod(𝑅, 𝑆) = (((1st ↾ (V × V)) ∘ (𝑅 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝑆 ∘ (2nd ↾ (V × V)))))
171, 16eqtr4i 2847 1 pprod(𝑅, 𝑆) = pprod(𝑅, 𝑆)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1533  Vcvv 3494   ∩ cin 3934   × cxp 5552  ◡ccnv 5553   ↾ cres 5556   ∘ ccom 5558  1st c1st 7686  2nd c2nd 7687  pprodcpprod 33292 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-br 5066  df-opab 5128  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-txp 33315  df-pprod 33316 This theorem is referenced by:  brpprod3b  33348
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