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Theorem ovmuc 6130
 Description: The value of cardinal multiplication. (Contributed by SF, 10-Mar-2015.)
Assertion
Ref Expression
ovmuc ((M NC N NC ) → (M ·c N) = {a b M g N a ≈ (b × g)})
Distinct variable groups:   a,b,g   M,a,b   N,a,b,g
Allowed substitution hint:   M(g)

Proof of Theorem ovmuc
Dummy variables c m n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elima 4754 . . . . 5 (a ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) “ M) ↔ b M b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N)a)
2 df-br 4640 . . . . . . 7 (b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N)ab, a (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N))
3 elima 4754 . . . . . . 7 (b, a (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) ↔ g N gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ )b, a)
4 df-br 4640 . . . . . . . . 9 (gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ )b, ag, b, a ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ))
5 elrn2 4897 . . . . . . . . . 10 (g, b, a ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) ↔ cc, g, b, a ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ))
6 elin 3219 . . . . . . . . . . . 12 (c, g, b, a ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) ↔ (c, g, b, a Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) c, g, b, a Ins2 Ins2 ≈ ))
7 vex 2862 . . . . . . . . . . . . . . 15 a V
87oqelins4 5794 . . . . . . . . . . . . . 14 (c, g, b, a Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ c, g, b ran ( Cross ⊗ (2nd ⊗ 1st )))
9 elrn 4896 . . . . . . . . . . . . . . 15 (c, g, b ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ a a( Cross ⊗ (2nd ⊗ 1st ))c, g, b)
10 trtxp 5781 . . . . . . . . . . . . . . . . 17 (a( Cross ⊗ (2nd ⊗ 1st ))c, g, b ↔ (a Cross c a(2nd ⊗ 1st )g, b))
11 trtxp 5781 . . . . . . . . . . . . . . . . . . 19 (a(2nd ⊗ 1st )g, b ↔ (a2nd g a1st b))
12 ancom 437 . . . . . . . . . . . . . . . . . . 19 ((a2nd g a1st b) ↔ (a1st b a2nd g))
13 vex 2862 . . . . . . . . . . . . . . . . . . . 20 b V
14 vex 2862 . . . . . . . . . . . . . . . . . . . 20 g V
1513, 14op1st2nd 5790 . . . . . . . . . . . . . . . . . . 19 ((a1st b a2nd g) ↔ a = b, g)
1611, 12, 153bitri 262 . . . . . . . . . . . . . . . . . 18 (a(2nd ⊗ 1st )g, ba = b, g)
1716anbi2i 675 . . . . . . . . . . . . . . . . 17 ((a Cross c a(2nd ⊗ 1st )g, b) ↔ (a Cross c a = b, g))
18 ancom 437 . . . . . . . . . . . . . . . . 17 ((a Cross c a = b, g) ↔ (a = b, g a Cross c))
1910, 17, 183bitri 262 . . . . . . . . . . . . . . . 16 (a( Cross ⊗ (2nd ⊗ 1st ))c, g, b ↔ (a = b, g a Cross c))
2019exbii 1582 . . . . . . . . . . . . . . 15 (a a( Cross ⊗ (2nd ⊗ 1st ))c, g, ba(a = b, g a Cross c))
2113, 14opex 4588 . . . . . . . . . . . . . . . 16 b, g V
22 breq1 4642 . . . . . . . . . . . . . . . 16 (a = b, g → (a Cross cb, g Cross c))
2321, 22ceqsexv 2894 . . . . . . . . . . . . . . 15 (a(a = b, g a Cross c) ↔ b, g Cross c)
249, 20, 233bitri 262 . . . . . . . . . . . . . 14 (c, g, b ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ b, g Cross c)
2513, 14brcross 5849 . . . . . . . . . . . . . 14 (b, g Cross cc = (b × g))
268, 24, 253bitri 262 . . . . . . . . . . . . 13 (c, g, b, a Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ↔ c = (b × g))
2714otelins2 5791 . . . . . . . . . . . . . 14 (c, g, b, a Ins2 Ins2 ≈ ↔ c, b, a Ins2 ≈ )
2813otelins2 5791 . . . . . . . . . . . . . 14 (c, b, a Ins2 ≈ ↔ c, a ≈ )
29 df-br 4640 . . . . . . . . . . . . . . 15 (cac, a ≈ )
30 brcnv 4892 . . . . . . . . . . . . . . 15 (caac)
3129, 30bitr3i 242 . . . . . . . . . . . . . 14 (c, a ≈ ↔ ac)
3227, 28, 313bitri 262 . . . . . . . . . . . . 13 (c, g, b, a Ins2 Ins2 ≈ ↔ ac)
3326, 32anbi12i 678 . . . . . . . . . . . 12 ((c, g, b, a Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) c, g, b, a Ins2 Ins2 ≈ ) ↔ (c = (b × g) ac))
346, 33bitri 240 . . . . . . . . . . 11 (c, g, b, a ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) ↔ (c = (b × g) ac))
3534exbii 1582 . . . . . . . . . 10 (cc, g, b, a ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) ↔ c(c = (b × g) ac))
3613, 14xpex 5115 . . . . . . . . . . 11 (b × g) V
37 breq2 4643 . . . . . . . . . . 11 (c = (b × g) → (aca ≈ (b × g)))
3836, 37ceqsexv 2894 . . . . . . . . . 10 (c(c = (b × g) ac) ↔ a ≈ (b × g))
395, 35, 383bitri 262 . . . . . . . . 9 (g, b, a ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) ↔ a ≈ (b × g))
404, 39bitri 240 . . . . . . . 8 (gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ )b, aa ≈ (b × g))
4140rexbii 2639 . . . . . . 7 (g N gran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ )b, ag N a ≈ (b × g))
422, 3, 413bitri 262 . . . . . 6 (b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N)ag N a ≈ (b × g))
4342rexbii 2639 . . . . 5 (b M b(ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N)ab M g N a ≈ (b × g))
441, 43bitri 240 . . . 4 (a ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) “ M) ↔ b M g N a ≈ (b × g))
4544abbi2i 2464 . . 3 ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) “ M) = {a b M g N a ≈ (b × g)}
46 crossex 5850 . . . . . . . . . . 11 Cross V
47 2ndex 5112 . . . . . . . . . . . 12 2nd V
48 1stex 4739 . . . . . . . . . . . 12 1st V
4947, 48txpex 5785 . . . . . . . . . . 11 (2nd ⊗ 1st ) V
5046, 49txpex 5785 . . . . . . . . . 10 ( Cross ⊗ (2nd ⊗ 1st )) V
5150rnex 5107 . . . . . . . . 9 ran ( Cross ⊗ (2nd ⊗ 1st )) V
5251ins4ex 5799 . . . . . . . 8 Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) V
53 enex 6031 . . . . . . . . . . 11 V
5453cnvex 5102 . . . . . . . . . 10 V
5554ins2ex 5797 . . . . . . . . 9 Ins2 V
5655ins2ex 5797 . . . . . . . 8 Ins2 Ins2 V
5752, 56inex 4105 . . . . . . 7 ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) V
5857rnex 5107 . . . . . 6 ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) V
59 imaexg 4746 . . . . . 6 ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) V N NC ) → (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) V)
6058, 59mpan 651 . . . . 5 (N NC → (ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) V)
61 imaexg 4746 . . . . 5 (((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) V M NC ) → ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) “ M) V)
6260, 61sylan 457 . . . 4 ((N NC M NC ) → ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) “ M) V)
6362ancoms 439 . . 3 ((M NC N NC ) → ((ran ( Ins4 ran ( Cross ⊗ (2nd ⊗ 1st )) ∩ Ins2 Ins2 ≈ ) “ N) “ M) V)
6445, 63syl5eqelr 2438 . 2 ((M NC N NC ) → {a b M g N a ≈ (b × g)} V)
65 rexeq 2808 . . . 4 (m = M → (b m g n a ≈ (b × g) ↔ b M g n a ≈ (b × g)))
6665abbidv 2467 . . 3 (m = M → {a b m g n a ≈ (b × g)} = {a b M g n a ≈ (b × g)})
67 rexeq 2808 . . . . 5 (n = N → (g n a ≈ (b × g) ↔ g N a ≈ (b × g)))
6867rexbidv 2635 . . . 4 (n = N → (b M g n a ≈ (b × g) ↔ b M g N a ≈ (b × g)))
6968abbidv 2467 . . 3 (n = N → {a b M g n a ≈ (b × g)} = {a b M g N a ≈ (b × g)})
70 df-muc 6102 . . 3 ·c = (m NC , n NC {a b m g n a ≈ (b × g)})
7166, 69, 70ovmpt2g 5715 . 2 ((M NC N NC {a b M g N a ≈ (b × g)} V) → (M ·c N) = {a b M g N a ≈ (b × g)})
7264, 71mpd3an3 1278 1 ((M NC N NC ) → (M ·c N) = {a b M g N a ≈ (b × g)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2615  Vcvv 2859   ∩ cin 3208  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   “ cima 4722   × cxp 4770  ◡ccnv 4771  ran crn 4773  2nd c2nd 4783  (class class class)co 5525   ⊗ ctxp 5735   Ins2 cins2 5749   Ins4 cins4 5755   Cross ccross 5763   ≈ cen 6028   NC cncs 6088   ·c cmuc 6092 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-csb 3137  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-iun 3971  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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-cross 5764  df-en 6029  df-muc 6102 This theorem is referenced by:  mucnc  6131
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