Theorem mucass 6136

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.

Step	Hyp	Ref	Expression
1		elncs 6120	. . . 4 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
2		elncs 6120	. . . 4 ⊢ (B ∈ NC ↔ ∃y B = Nc y)
3		elncs 6120	. . . 4 ⊢ (C ∈ NC ↔ ∃z C = Nc z)
4	1, 2, 3	3anbi123i 1140	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ↔ (∃x A = Nc x ∧ ∃y B = Nc y ∧ ∃z C = Nc z))
5		eeeanv 1914	. . 3 ⊢ (∃x∃y∃z(A = Nc x ∧ B = Nc y ∧ C = Nc z) ↔ (∃x A = Nc x ∧ ∃y B = Nc y ∧ ∃z C = Nc z))
6	4, 5	bitr4i 243	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) ↔ ∃x∃y∃z(A = Nc x ∧ B = Nc y ∧ C = Nc z))
7		vex 2863	. . . . . . . 8 ⊢ x ∈ V
8		vex 2863	. . . . . . . 8 ⊢ y ∈ V
9		vex 2863	. . . . . . . 8 ⊢ z ∈ V
10	7, 8, 9	xpassen 6058	. . . . . . 7 ⊢ ((x × y) × z) ≈ (x × (y × z))
11	7, 8	xpex 5116	. . . . . . . . 9 ⊢ (x × y) ∈ V
12	11, 9	xpex 5116	. . . . . . . 8 ⊢ ((x × y) × z) ∈ V
13	12	eqnc 6128	. . . . . . 7 ⊢ ( Nc ((x × y) × z) = Nc (x × (y × z)) ↔ ((x × y) × z) ≈ (x × (y × z)))
14	10, 13	mpbir 200	. . . . . 6 ⊢ Nc ((x × y) × z) = Nc (x × (y × z))
15	7, 8	mucnc 6132	. . . . . . . 8 ⊢ ( Nc x ·c Nc y) = Nc (x × y)
16	15	oveq1i 5534	. . . . . . 7 ⊢ (( Nc x ·c Nc y) ·c Nc z) = ( Nc (x × y) ·c Nc z)
17	11, 9	mucnc 6132	. . . . . . 7 ⊢ ( Nc (x × y) ·c Nc z) = Nc ((x × y) × z)
18	16, 17	eqtri 2373	. . . . . 6 ⊢ (( Nc x ·c Nc y) ·c Nc z) = Nc ((x × y) × z)
19	8, 9	mucnc 6132	. . . . . . . 8 ⊢ ( Nc y ·c Nc z) = Nc (y × z)
20	19	oveq2i 5535	. . . . . . 7 ⊢ ( Nc x ·c ( Nc y ·c Nc z)) = ( Nc x ·c Nc (y × z))
21	8, 9	xpex 5116	. . . . . . . 8 ⊢ (y × z) ∈ V
22	7, 21	mucnc 6132	. . . . . . 7 ⊢ ( Nc x ·c Nc (y × z)) = Nc (x × (y × z))
23	20, 22	eqtri 2373	. . . . . 6 ⊢ ( Nc x ·c ( Nc y ·c Nc z)) = Nc (x × (y × z))
24	14, 18, 23	3eqtr4i 2383	. . . . 5 ⊢ (( Nc x ·c Nc y) ·c Nc z) = ( Nc x ·c ( Nc y ·c Nc z))
25		oveq12 5533	. . . . . . 7 ⊢ ((A = Nc x ∧ B = Nc y) → (A ·c B) = ( Nc x ·c Nc y))
26		id 19	. . . . . . 7 ⊢ (C = Nc z → C = Nc z)
27	25, 26	oveqan12d 5542	. . . . . 6 ⊢ (((A = Nc x ∧ B = Nc y) ∧ C = Nc z) → ((A ·c B) ·c C) = (( Nc x ·c Nc y) ·c Nc z))
28	27	3impa 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 x → A = Nc x)
30		oveq12 5533	. . . . . . 7 ⊢ ((B = Nc y ∧ C = Nc z) → (B ·c C) = ( Nc y ·c Nc z))
31	29, 30	oveqan12d 5542	. . . . . 6 ⊢ ((A = Nc x ∧ (B = Nc y ∧ C = Nc z)) → (A ·c (B ·c C)) = ( Nc x ·c ( Nc y ·c Nc z)))
32	31	3impb 1147	. . . . 5 ⊢ ((A = Nc x ∧ B = Nc y ∧ C = Nc z) → (A ·c (B ·c C)) = ( Nc x ·c ( Nc y ·c Nc z)))
33	24, 28, 32	3eqtr4a 2411	. . . 4 ⊢ ((A = Nc x ∧ B = Nc y ∧ C = Nc z) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
34	33	exlimiv 1634	. . 3 ⊢ (∃z(A = Nc x ∧ B = Nc y ∧ C = Nc z) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
35	34	exlimivv 1635	. 2 ⊢ (∃x∃y∃z(A = Nc x ∧ B = Nc y ∧ C = Nc z) → ((A ·c B) ·c C) = (A ·c (B ·c C)))
36	6, 35	sylbi 187	1 ⊢ ((A ∈ NC ∧ B ∈ NC ∧ C ∈ NC ) → ((A ·c B) ·c C) = (A ·c (B ·c C)))

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710   class class class wbr 4640   × cxp 4771  (class class class)co 5526   ≈ cen 6029   NC cncs 6089   Nc cnc 6092   ·_c cmuc 6093

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-csb 3138  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-iun 3972  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-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-pprod 5739  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-cross 5765  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-nc 6102  df-muc 6103

This theorem is referenced by: (None)