Theorem muccom 6134

Description: Cardinal multiplication commutes. Theorem XI.2.28 of [Rosser] p. 378. (Contributed by SF, 10-Mar-2015.)

Assertion

Ref	Expression
muccom	⊢ ((A ∈ NC ∧ B ∈ NC ) → (A ·c B) = (B ·c A))

Proof of Theorem muccom

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elncs 6119	. . . 4 ⊢ (A ∈ NC ↔ ∃x A = Nc x)
2		elncs 6119	. . . 4 ⊢ (B ∈ NC ↔ ∃y B = Nc y)
3	1, 2	anbi12i 678	. . 3 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
4		eeanv 1913	. . 3 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) ↔ (∃x A = Nc x ∧ ∃y B = Nc y))
5	3, 4	bitr4i 243	. 2 ⊢ ((A ∈ NC ∧ B ∈ NC ) ↔ ∃x∃y(A = Nc x ∧ B = Nc y))
6		vex 2862	. . . . . . 7 ⊢ x ∈ V
7		vex 2862	. . . . . . 7 ⊢ y ∈ V
8	6, 7	xpcomen 6052	. . . . . 6 ⊢ (x × y) ≈ (y × x)
9	6, 7	xpex 5115	. . . . . . 7 ⊢ (x × y) ∈ V
10	9	eqnc 6127	. . . . . 6 ⊢ ( Nc (x × y) = Nc (y × x) ↔ (x × y) ≈ (y × x))
11	8, 10	mpbir 200	. . . . 5 ⊢ Nc (x × y) = Nc (y × x)
12	6, 7	mucnc 6131	. . . . 5 ⊢ ( Nc x ·c Nc y) = Nc (x × y)
13	7, 6	mucnc 6131	. . . . 5 ⊢ ( Nc y ·c Nc x) = Nc (y × x)
14	11, 12, 13	3eqtr4i 2383	. . . 4 ⊢ ( Nc x ·c Nc y) = ( Nc y ·c Nc x)
15		oveq12 5532	. . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → (A ·c B) = ( Nc x ·c Nc y))
16		oveq12 5532	. . . . 5 ⊢ ((B = Nc y ∧ A = Nc x) → (B ·c A) = ( Nc y ·c Nc x))
17	16	ancoms 439	. . . 4 ⊢ ((A = Nc x ∧ B = Nc y) → (B ·c A) = ( Nc y ·c Nc x))
18	14, 15, 17	3eqtr4a 2411	. . 3 ⊢ ((A = Nc x ∧ B = Nc y) → (A ·c B) = (B ·c A))
19	18	exlimivv 1635	. 2 ⊢ (∃x∃y(A = Nc x ∧ B = Nc y) → (A ·c B) = (B ·c A))
20	5, 19	sylbi 187	1 ⊢ ((A ∈ NC ∧ B ∈ NC ) → (A ·c B) = (B ·c A))

Syntax hints:   → wi 4   ∧ wa 358  ∃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-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-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:  addcdir  6251  muc0or  6252  lemuc2  6254