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Theorem cnvpprod 5841
 Description: The converse of a parallel product. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
cnvpprod PProd (A, B) = PProd (A, B)

Proof of Theorem cnvpprod
StepHypRef Expression
1 cnvin 5035 . . 3 ((1st (A 1st )) ∩ (2nd (B 2nd ))) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
2 cnvco 4894 . . . . 5 (1st (A 1st )) = ((A 1st ) 1st )
3 cnvco 4894 . . . . . 6 (A 1st ) = (1st A)
4 cnvcnv 5062 . . . . . 6 1st = 1st
53, 4coeq12i 4880 . . . . 5 ((A 1st ) 1st ) = ((1st A) 1st )
6 coass 5097 . . . . 5 ((1st A) 1st ) = (1st (A 1st ))
72, 5, 63eqtri 2377 . . . 4 (1st (A 1st )) = (1st (A 1st ))
8 cnvco 4894 . . . . 5 (2nd (B 2nd )) = ((B 2nd ) 2nd )
9 cnvco 4894 . . . . . 6 (B 2nd ) = (2nd B)
10 cnvcnv 5062 . . . . . 6 2nd = 2nd
119, 10coeq12i 4880 . . . . 5 ((B 2nd ) 2nd ) = ((2nd B) 2nd )
12 coass 5097 . . . . 5 ((2nd B) 2nd ) = (2nd (B 2nd ))
138, 11, 123eqtri 2377 . . . 4 (2nd (B 2nd )) = (2nd (B 2nd ))
147, 13ineq12i 3455 . . 3 ((1st (A 1st )) ∩ (2nd (B 2nd ))) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
151, 14eqtri 2373 . 2 ((1st (A 1st )) ∩ (2nd (B 2nd ))) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
16 df-pprod 5738 . . . 4 PProd (A, B) = ((A 1st ) ⊗ (B 2nd ))
17 df-txp 5736 . . . 4 ((A 1st ) ⊗ (B 2nd )) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
1816, 17eqtri 2373 . . 3 PProd (A, B) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
1918cnveqi 4887 . 2 PProd (A, B) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
20 df-pprod 5738 . . 3 PProd (A, B) = ((A 1st ) ⊗ (B 2nd ))
21 df-txp 5736 . . 3 ((A 1st ) ⊗ (B 2nd )) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
2220, 21eqtri 2373 . 2 PProd (A, B) = ((1st (A 1st )) ∩ (2nd (B 2nd )))
2315, 19, 223eqtr4i 2383 1 PProd (A, B) = PProd (A, B)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∩ cin 3208  1st c1st 4717   ∘ ccom 4721  ◡ccnv 4771  2nd c2nd 4783   ⊗ ctxp 5735   PProd cpprod 5737 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-cnv 4785  df-txp 5736  df-pprod 5738 This theorem is referenced by:  rnpprod  5842  f1opprod  5844
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