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Theorem mucass 6135
 Description: Cardinal multiplication associates. Theorem XI.2.29 of [Rosser] p. 378. (Contributed by SF, 10-Mar-2015.)
Assertion
Ref Expression
mucass ((A NC B NC C NC ) → ((A ·c B) ·c C) = (A ·c (B ·c C)))

Proof of Theorem mucass
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elncs 6119 . . . 4 (A NCx A = Nc x)
2 elncs 6119 . . . 4 (B NCy B = Nc y)
3 elncs 6119 . . . 4 (C NCz C = Nc z)
41, 2, 33anbi123i 1140 . . 3 ((A NC B NC C NC ) ↔ (x A = Nc x y B = Nc y z C = Nc z))
5 eeeanv 1914 . . 3 (xyz(A = Nc x B = Nc y C = Nc z) ↔ (x A = Nc x y B = Nc y z C = Nc z))
64, 5bitr4i 243 . 2 ((A NC B NC C NC ) ↔ xyz(A = Nc x B = Nc y C = Nc z))
7 vex 2862 . . . . . . . 8 x V
8 vex 2862 . . . . . . . 8 y V
9 vex 2862 . . . . . . . 8 z V
107, 8, 9xpassen 6057 . . . . . . 7 ((x × y) × z) ≈ (x × (y × z))
117, 8xpex 5115 . . . . . . . . 9 (x × y) V
1211, 9xpex 5115 . . . . . . . 8 ((x × y) × z) V
1312eqnc 6127 . . . . . . 7 ( Nc ((x × y) × z) = Nc (x × (y × z)) ↔ ((x × y) × z) ≈ (x × (y × z)))
1410, 13mpbir 200 . . . . . 6 Nc ((x × y) × z) = Nc (x × (y × z))
157, 8mucnc 6131 . . . . . . . 8 ( Nc x ·c Nc y) = Nc (x × y)
1615oveq1i 5533 . . . . . . 7 (( Nc x ·c Nc y) ·c Nc z) = ( Nc (x × y) ·c Nc z)
1711, 9mucnc 6131 . . . . . . 7 ( Nc (x × y) ·c Nc z) = Nc ((x × y) × z)
1816, 17eqtri 2373 . . . . . 6 (( Nc x ·c Nc y) ·c Nc z) = Nc ((x × y) × z)
198, 9mucnc 6131 . . . . . . . 8 ( Nc y ·c Nc z) = Nc (y × z)
2019oveq2i 5534 . . . . . . 7 ( Nc x ·c ( Nc y ·c Nc z)) = ( Nc x ·c Nc (y × z))
218, 9xpex 5115 . . . . . . . 8 (y × z) V
227, 21mucnc 6131 . . . . . . 7 ( Nc x ·c Nc (y × z)) = Nc (x × (y × z))
2320, 22eqtri 2373 . . . . . 6 ( Nc x ·c ( Nc y ·c Nc z)) = Nc (x × (y × z))
2414, 18, 233eqtr4i 2383 . . . . 5 (( Nc x ·c Nc y) ·c Nc z) = ( Nc x ·c ( Nc y ·c Nc z))
25 oveq12 5532 . . . . . . 7 ((A = Nc x B = Nc y) → (A ·c B) = ( Nc x ·c Nc y))
26 id 19 . . . . . . 7 (C = Nc zC = Nc z)
2725, 26oveqan12d 5541 . . . . . 6 (((A = Nc x B = Nc y) C = Nc z) → ((A ·c B) ·c C) = (( Nc x ·c Nc y) ·c Nc z))
28273impa 1146 . . . . 5 ((A = Nc x B = Nc y C = Nc z) → ((A ·c B) ·c C) = (( Nc x ·c Nc y) ·c Nc z))
29 id 19 . . . . . . 7 (A = Nc xA = Nc x)
30 oveq12 5532 . . . . . . 7 ((B = Nc y C = Nc z) → (B ·c C) = ( Nc y ·c Nc z))
3129, 30oveqan12d 5541 . . . . . 6 ((A = Nc x (B = Nc y C = Nc z)) → (A ·c (B ·c C)) = ( Nc x ·c ( Nc y ·c Nc z)))
32313impb 1147 . . . . 5 ((A = Nc x B = Nc y C = Nc z) → (A ·c (B ·c C)) = ( Nc x ·c ( Nc y ·c Nc z)))
3324, 28, 323eqtr4a 2411 . . . 4 ((A = Nc x B = Nc y C = Nc z) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
3433exlimiv 1634 . . 3 (z(A = Nc x B = Nc y C = Nc z) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
3534exlimivv 1635 . 2 (xyz(A = Nc x B = Nc y C = Nc z) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
366, 35sylbi 187 1 ((A NC B NC C NC ) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   class class class wbr 4639   × cxp 4770  (class class class)co 5525   ≈ cen 6028   NC cncs 6088   Nc cnc 6091   ·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-pprod 5738  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-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-en 6029  df-ncs 6098  df-nc 6101  df-muc 6102 This theorem is referenced by: (None)
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