Theorem cnvpprod 5842

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

Step	Hyp	Ref	Expression
1		cnvin 5036	. . 3 ⊢ ◡((◡1st ∘ (A ∘ 1st )) ∩ (◡2nd ∘ (B ∘ 2nd ))) = (◡(◡1st ∘ (A ∘ 1st )) ∩ ◡(◡2nd ∘ (B ∘ 2nd )))
2		cnvco 4895	. . . . 5 ⊢ ◡(◡1st ∘ (A ∘ 1st )) = (◡(A ∘ 1st ) ∘ ◡◡1st )
3		cnvco 4895	. . . . . 6 ⊢ ◡(A ∘ 1st ) = (◡1st ∘ ◡A)
4		cnvcnv 5063	. . . . . 6 ⊢ ◡◡1st = 1st
5	3, 4	coeq12i 4881	. . . . 5 ⊢ (◡(A ∘ 1st ) ∘ ◡◡1st ) = ((◡1st ∘ ◡A) ∘ 1st )
6		coass 5098	. . . . 5 ⊢ ((◡1st ∘ ◡A) ∘ 1st ) = (◡1st ∘ (◡A ∘ 1st ))
7	2, 5, 6	3eqtri 2377	. . . 4 ⊢ ◡(◡1st ∘ (A ∘ 1st )) = (◡1st ∘ (◡A ∘ 1st ))
8		cnvco 4895	. . . . 5 ⊢ ◡(◡2nd ∘ (B ∘ 2nd )) = (◡(B ∘ 2nd ) ∘ ◡◡2nd )
9		cnvco 4895	. . . . . 6 ⊢ ◡(B ∘ 2nd ) = (◡2nd ∘ ◡B)
10		cnvcnv 5063	. . . . . 6 ⊢ ◡◡2nd = 2nd
11	9, 10	coeq12i 4881	. . . . 5 ⊢ (◡(B ∘ 2nd ) ∘ ◡◡2nd ) = ((◡2nd ∘ ◡B) ∘ 2nd )
12		coass 5098	. . . . 5 ⊢ ((◡2nd ∘ ◡B) ∘ 2nd ) = (◡2nd ∘ (◡B ∘ 2nd ))
13	8, 11, 12	3eqtri 2377	. . . 4 ⊢ ◡(◡2nd ∘ (B ∘ 2nd )) = (◡2nd ∘ (◡B ∘ 2nd ))
14	7, 13	ineq12i 3456	. . 3 ⊢ (◡(◡1st ∘ (A ∘ 1st )) ∩ ◡(◡2nd ∘ (B ∘ 2nd ))) = ((◡1st ∘ (◡A ∘ 1st )) ∩ (◡2nd ∘ (◡B ∘ 2nd )))
15	1, 14	eqtri 2373	. 2 ⊢ ◡((◡1st ∘ (A ∘ 1st )) ∩ (◡2nd ∘ (B ∘ 2nd ))) = ((◡1st ∘ (◡A ∘ 1st )) ∩ (◡2nd ∘ (◡B ∘ 2nd )))
16		df-pprod 5739	. . . 4 ⊢ PProd (A, B) = ((A ∘ 1st ) ⊗ (B ∘ 2nd ))
17		df-txp 5737	. . . 4 ⊢ ((A ∘ 1st ) ⊗ (B ∘ 2nd )) = ((◡1st ∘ (A ∘ 1st )) ∩ (◡2nd ∘ (B ∘ 2nd )))
18	16, 17	eqtri 2373	. . 3 ⊢ PProd (A, B) = ((◡1st ∘ (A ∘ 1st )) ∩ (◡2nd ∘ (B ∘ 2nd )))
19	18	cnveqi 4888	. 2 ⊢ ◡ PProd (A, B) = ◡((◡1st ∘ (A ∘ 1st )) ∩ (◡2nd ∘ (B ∘ 2nd )))
20		df-pprod 5739	. . 3 ⊢ PProd (◡A, ◡B) = ((◡A ∘ 1st ) ⊗ (◡B ∘ 2nd ))
21		df-txp 5737	. . 3 ⊢ ((◡A ∘ 1st ) ⊗ (◡B ∘ 2nd )) = ((◡1st ∘ (◡A ∘ 1st )) ∩ (◡2nd ∘ (◡B ∘ 2nd )))
22	20, 21	eqtri 2373	. 2 ⊢ PProd (◡A, ◡B) = ((◡1st ∘ (◡A ∘ 1st )) ∩ (◡2nd ∘ (◡B ∘ 2nd )))
23	15, 19, 22	3eqtr4i 2383	1 ⊢ ◡ PProd (A, B) = PProd (◡A, ◡B)

Syntax hints:   = wceq 1642   ∩ cin 3209  1^st c1st 4718   ∘ ccom 4722  ^◡ccnv 4772  2^nd c2nd 4784   ⊗ ctxp 5736   PProd cpprod 5738

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-cnv 4786  df-txp 5737  df-pprod 5739

This theorem is referenced by:  rnpprod  5843  f1opprod  5845