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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-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-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-muc 6102 This theorem is referenced by: (None)
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