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Theorem mucex 6133
Description: Cardinal multiplication is a set. (Contributed by SF, 24-Feb-2015.)
Assertion
Ref Expression
mucex ·c V

Proof of Theorem mucex
Dummy variables a b c d m n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-muc 6102 . . 3 ·c = (m NC , n NC {a b m c n a ≈ (b × c)})
2 elin 3219 . . . . . . . . 9 ({c}, {d}, m, n ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) ↔ ({c}, {d}, m, n Ins2 Ins2 S {c}, {d}, m, n Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)))
3 snex 4111 . . . . . . . . . . . 12 {d} V
43otelins2 5791 . . . . . . . . . . 11 ({c}, {d}, m, n Ins2 Ins2 S {c}, m, n Ins2 S )
5 vex 2862 . . . . . . . . . . . 12 m V
65otelins2 5791 . . . . . . . . . . 11 ({c}, m, n Ins2 S {c}, n S )
7 vex 2862 . . . . . . . . . . . 12 c V
8 vex 2862 . . . . . . . . . . . 12 n V
97, 8opelssetsn 4760 . . . . . . . . . . 11 ({c}, n S c n)
104, 6, 93bitri 262 . . . . . . . . . 10 ({c}, {d}, m, n Ins2 Ins2 S c n)
118oqelins4 5794 . . . . . . . . . . 11 ({c}, {d}, m, n Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c) ↔ {c}, {d}, m (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c))
12 elin 3219 . . . . . . . . . . . . . 14 ({b}, {c}, {d}, m ( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) ↔ ({b}, {c}, {d}, m Ins2 Ins2 S {b}, {c}, {d}, m Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )))
13 snex 4111 . . . . . . . . . . . . . . . . 17 {c} V
1413otelins2 5791 . . . . . . . . . . . . . . . 16 ({b}, {c}, {d}, m Ins2 Ins2 S {b}, {d}, m Ins2 S )
153otelins2 5791 . . . . . . . . . . . . . . . 16 ({b}, {d}, m Ins2 S {b}, m S )
16 vex 2862 . . . . . . . . . . . . . . . . 17 b V
1716, 5opelssetsn 4760 . . . . . . . . . . . . . . . 16 ({b}, m S b m)
1814, 15, 173bitri 262 . . . . . . . . . . . . . . 15 ({b}, {c}, {d}, m Ins2 Ins2 S b m)
195oqelins4 5794 . . . . . . . . . . . . . . . 16 ({b}, {c}, {d}, m Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ ) ↔ {b}, {c}, {d} SI3 ran ( Ins4 CrossIns2 Ins2 ≈ ))
20 vex 2862 . . . . . . . . . . . . . . . . 17 d V
2116, 7, 20otsnelsi3 5805 . . . . . . . . . . . . . . . 16 ({b}, {c}, {d} SI3 ran ( Ins4 CrossIns2 Ins2 ≈ ) ↔ b, c, d ran ( Ins4 CrossIns2 Ins2 ≈ ))
22 elrn2 4897 . . . . . . . . . . . . . . . . 17 (b, c, d ran ( Ins4 CrossIns2 Ins2 ≈ ) ↔ aa, b, c, d ( Ins4 CrossIns2 Ins2 ≈ ))
23 elin 3219 . . . . . . . . . . . . . . . . . . 19 (a, b, c, d ( Ins4 CrossIns2 Ins2 ≈ ) ↔ (a, b, c, d Ins4 Cross a, b, c, d Ins2 Ins2 ≈ ))
2420oqelins4 5794 . . . . . . . . . . . . . . . . . . . . 21 (a, b, c, d Ins4 Crossa, b, c Cross )
25 df-br 4640 . . . . . . . . . . . . . . . . . . . . 21 (a Cross b, ca, b, c Cross )
26 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . 22 (a Cross b, cb, c Cross a)
2716, 7brcross 5849 . . . . . . . . . . . . . . . . . . . . . 22 (b, c Cross aa = (b × c))
2826, 27bitri 240 . . . . . . . . . . . . . . . . . . . . 21 (a Cross b, ca = (b × c))
2924, 25, 283bitr2i 264 . . . . . . . . . . . . . . . . . . . 20 (a, b, c, d Ins4 Crossa = (b × c))
3016otelins2 5791 . . . . . . . . . . . . . . . . . . . . 21 (a, b, c, d Ins2 Ins2 ≈ ↔ a, c, d Ins2 ≈ )
317otelins2 5791 . . . . . . . . . . . . . . . . . . . . . 22 (a, c, d Ins2 ≈ ↔ a, d ≈ )
32 df-br 4640 . . . . . . . . . . . . . . . . . . . . . 22 (ada, d ≈ )
33 brcnv 4892 . . . . . . . . . . . . . . . . . . . . . 22 (adda)
3431, 32, 333bitr2i 264 . . . . . . . . . . . . . . . . . . . . 21 (a, c, d Ins2 ≈ ↔ da)
3530, 34bitri 240 . . . . . . . . . . . . . . . . . . . 20 (a, b, c, d Ins2 Ins2 ≈ ↔ da)
3629, 35anbi12i 678 . . . . . . . . . . . . . . . . . . 19 ((a, b, c, d Ins4 Cross a, b, c, d Ins2 Ins2 ≈ ) ↔ (a = (b × c) da))
3723, 36bitri 240 . . . . . . . . . . . . . . . . . 18 (a, b, c, d ( Ins4 CrossIns2 Ins2 ≈ ) ↔ (a = (b × c) da))
3837exbii 1582 . . . . . . . . . . . . . . . . 17 (aa, b, c, d ( Ins4 CrossIns2 Ins2 ≈ ) ↔ a(a = (b × c) da))
3916, 7xpex 5115 . . . . . . . . . . . . . . . . . 18 (b × c) V
40 breq2 4643 . . . . . . . . . . . . . . . . . 18 (a = (b × c) → (dad ≈ (b × c)))
4139, 40ceqsexv 2894 . . . . . . . . . . . . . . . . 17 (a(a = (b × c) da) ↔ d ≈ (b × c))
4222, 38, 413bitri 262 . . . . . . . . . . . . . . . 16 (b, c, d ran ( Ins4 CrossIns2 Ins2 ≈ ) ↔ d ≈ (b × c))
4319, 21, 423bitri 262 . . . . . . . . . . . . . . 15 ({b}, {c}, {d}, m Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ ) ↔ d ≈ (b × c))
4418, 43anbi12i 678 . . . . . . . . . . . . . 14 (({b}, {c}, {d}, m Ins2 Ins2 S {b}, {c}, {d}, m Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) ↔ (b m d ≈ (b × c)))
4512, 44bitri 240 . . . . . . . . . . . . 13 ({b}, {c}, {d}, m ( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) ↔ (b m d ≈ (b × c)))
4645exbii 1582 . . . . . . . . . . . 12 (b{b}, {c}, {d}, m ( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) ↔ b(b m d ≈ (b × c)))
47 elima1c 4947 . . . . . . . . . . . 12 ({c}, {d}, m (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c) ↔ b{b}, {c}, {d}, m ( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )))
48 df-rex 2620 . . . . . . . . . . . 12 (b m d ≈ (b × c) ↔ b(b m d ≈ (b × c)))
4946, 47, 483bitr4i 268 . . . . . . . . . . 11 ({c}, {d}, m (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c) ↔ b m d ≈ (b × c))
5011, 49bitri 240 . . . . . . . . . 10 ({c}, {d}, m, n Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c) ↔ b m d ≈ (b × c))
5110, 50anbi12i 678 . . . . . . . . 9 (({c}, {d}, m, n Ins2 Ins2 S {c}, {d}, m, n Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) ↔ (c n b m d ≈ (b × c)))
522, 51bitri 240 . . . . . . . 8 ({c}, {d}, m, n ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) ↔ (c n b m d ≈ (b × c)))
5352exbii 1582 . . . . . . 7 (c{c}, {d}, m, n ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) ↔ c(c n b m d ≈ (b × c)))
54 df-rex 2620 . . . . . . 7 (c n b m d ≈ (b × c) ↔ c(c n b m d ≈ (b × c)))
5553, 54bitr4i 243 . . . . . 6 (c{c}, {d}, m, n ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) ↔ c n b m d ≈ (b × c))
56 elima1c 4947 . . . . . 6 ({d}, m, n (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c) ↔ c{c}, {d}, m, n ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)))
57 rexcom 2772 . . . . . 6 (b m c n d ≈ (b × c) ↔ c n b m d ≈ (b × c))
5855, 56, 573bitr4i 268 . . . . 5 ({d}, m, n (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c) ↔ b m c n d ≈ (b × c))
59 breq1 4642 . . . . . . 7 (a = d → (a ≈ (b × c) ↔ d ≈ (b × c)))
60592rexbidv 2657 . . . . . 6 (a = d → (b m c n a ≈ (b × c) ↔ b m c n d ≈ (b × c)))
6120, 60elab 2985 . . . . 5 (d {a b m c n a ≈ (b × c)} ↔ b m c n d ≈ (b × c))
6258, 61bitr4i 243 . . . 4 ({d}, m, n (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c) ↔ d {a b m c n a ≈ (b × c)})
6362releqmpt2 5809 . . 3 ((( NC × NC ) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c)) “ 1c)) = (m NC , n NC {a b m c n a ≈ (b × c)})
641, 63eqtr4i 2376 . 2 ·c = ((( NC × NC ) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c)) “ 1c))
65 ncsex 6111 . . 3 NC V
66 ssetex 4744 . . . . . . 7 S V
6766ins2ex 5797 . . . . . 6 Ins2 S V
6867ins2ex 5797 . . . . 5 Ins2 Ins2 S V
69 crossex 5850 . . . . . . . . . . . . . 14 Cross V
7069cnvex 5102 . . . . . . . . . . . . 13 Cross V
7170ins4ex 5799 . . . . . . . . . . . 12 Ins4 Cross V
72 enex 6031 . . . . . . . . . . . . . . 15 V
7372cnvex 5102 . . . . . . . . . . . . . 14 V
7473ins2ex 5797 . . . . . . . . . . . . 13 Ins2 V
7574ins2ex 5797 . . . . . . . . . . . 12 Ins2 Ins2 V
7671, 75inex 4105 . . . . . . . . . . 11 ( Ins4 CrossIns2 Ins2 ≈ ) V
7776rnex 5107 . . . . . . . . . 10 ran ( Ins4 CrossIns2 Ins2 ≈ ) V
7877si3ex 5806 . . . . . . . . 9 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ ) V
7978ins4ex 5799 . . . . . . . 8 Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ ) V
8068, 79inex 4105 . . . . . . 7 ( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) V
81 1cex 4142 . . . . . . 7 1c V
8280, 81imaex 4747 . . . . . 6 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c) V
8382ins4ex 5799 . . . . 5 Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c) V
8468, 83inex 4105 . . . 4 ( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) V
8584, 81imaex 4747 . . 3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c) V
8665, 65, 85mpt2exlem 5811 . 2 ((( NC × NC ) × V) (( Ins2 S Ins3 (( Ins2 Ins2 S Ins4 (( Ins2 Ins2 S Ins4 SI3 ran ( Ins4 CrossIns2 Ins2 ≈ )) “ 1c)) “ 1c)) “ 1c)) V
8764, 86eqeltri 2423 1 ·c V
Colors of variables: wff setvar class
Syntax hints:   wa 358  wex 1541   = wceq 1642   wcel 1710  {cab 2339  wrex 2615  Vcvv 2859   cdif 3206  cin 3208  csymdif 3209  {csn 3737  1cc1c 4134  cop 4561   class class class wbr 4639   S csset 4719  cima 4722   × cxp 4770  ccnv 4771  ran crn 4773   cmpt2 5653   Ins2 cins2 5749   Ins3 cins3 5751   Ins4 cins4 5755   SI3 csi3 5757   Cross ccross 5763  cen 6028   NC cncs 6088   ·c cmuc 6092
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-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-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-muc 6102
This theorem is referenced by: (None)
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