Theorem mucnc 6131

Description: Cardinal multiplication in terms of cardinality. Theorem XI.2.27 of [Rosser] p. 378. (Contributed by SF, 10-Mar-2015.)

Hypotheses

Ref	Expression
mucnc.1	|- A e. _V
mucnc.2	|- B e. _V

Assertion

Ref	Expression
mucnc	|- ( Nc A ·c Nc B) = Nc (A X. B)

Proof of Theorem mucnc

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

Step	Hyp	Ref	Expression
1		mucnc.1	. . . 4 |- A e. _V
2	1	ncelncsi 6121	. . 3 |- Nc A e. NC
3		mucnc.2	. . . 4 |- B e. _V
4	3	ncelncsi 6121	. . 3 |- Nc B e. NC
5		ovmuc 6130	. . 3 |- (( Nc A e. NC /\ Nc B e. NC ) -> ( Nc A ·c Nc B) = {x | E.y e. Nc AE.z e. Nc Bx ~~ (y X. z)})
6	2, 4, 5	mp2an 653	. 2 |- ( Nc A ·c Nc B) = {x | E.y e. Nc AE.z e. Nc Bx ~~ (y X. z)}
7		df-nc 6101	. . 3 |- Nc (A X. B) = [(A X. B)] ~~
8		dfec2 5948	. . 3 |- [(A X. B)] ~~ = {x | (A X. B) ~~ x}
9		elnc 6125	. . . . . . . 8 |- (y e. Nc A <-> y ~~ A)
10		elnc 6125	. . . . . . . 8 |- (z e. Nc B <-> z ~~ B)
11	9, 10	anbi12i 678	. . . . . . 7 |- ((y e. Nc A /\ z e. Nc B) <-> (y ~~ A /\ z ~~ B))
12		ensym 6037	. . . . . . 7 |- (x ~~ (y X. z) <-> (y X. z) ~~ x)
13	11, 12	anbi12i 678	. . . . . 6 |- (((y e. Nc A /\ z e. Nc B) /\ x ~~ (y X. z)) <-> ((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x))
14	13	2exbii 1583	. . . . 5 |- (E.yE.z((y e. Nc A /\ z e. Nc B) /\ x ~~ (y X. z)) <-> E.yE.z((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x))
15		r2ex 2652	. . . . 5 |- (E.y e. Nc AE.z e. Nc Bx ~~ (y X. z) <-> E.yE.z((y e. Nc A /\ z e. Nc B) /\ x ~~ (y X. z)))
16	1	enrflx 6035	. . . . . . 7 |- A ~~ A
17	3	enrflx 6035	. . . . . . 7 |- B ~~ B
18		breq1 4642	. . . . . . . . . 10 |- (y = A -> (y ~~ A <-> A ~~ A))
19		breq1 4642	. . . . . . . . . 10 |- (z = B -> (z ~~ B <-> B ~~ B))
20	18, 19	bi2anan9 843	. . . . . . . . 9 |- ((y = A /\ z = B) -> ((y ~~ A /\ z ~~ B) <-> (A ~~ A /\ B ~~ B)))
21		xpeq12 4803	. . . . . . . . . 10 |- ((y = A /\ z = B) -> (y X. z) = (A X. B))
22	21	breq1d 4649	. . . . . . . . 9 |- ((y = A /\ z = B) -> ((y X. z) ~~ x <-> (A X. B) ~~ x))
23	20, 22	anbi12d 691	. . . . . . . 8 |- ((y = A /\ z = B) -> (((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x) <-> ((A ~~ A /\ B ~~ B) /\ (A X. B) ~~ x)))
24	1, 3, 23	spc2ev 2947	. . . . . . 7 |- (((A ~~ A /\ B ~~ B) /\ (A X. B) ~~ x) -> E.yE.z((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x))
25	16, 17, 24	mpanl12 663	. . . . . 6 |- ((A X. B) ~~ x -> E.yE.z((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x))
26		xpen 6055	. . . . . . . . 9 |- ((y ~~ A /\ z ~~ B) -> (y X. z) ~~ (A X. B))
27		ensym 6037	. . . . . . . . 9 |- ((y X. z) ~~ (A X. B) <-> (A X. B) ~~ (y X. z))
28	26, 27	sylib 188	. . . . . . . 8 |- ((y ~~ A /\ z ~~ B) -> (A X. B) ~~ (y X. z))
29		entr 6038	. . . . . . . 8 |- (((A X. B) ~~ (y X. z) /\ (y X. z) ~~ x) -> (A X. B) ~~ x)
30	28, 29	sylan 457	. . . . . . 7 |- (((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x) -> (A X. B) ~~ x)
31	30	exlimivv 1635	. . . . . 6 |- (E.yE.z((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x) -> (A X. B) ~~ x)
32	25, 31	impbii 180	. . . . 5 |- ((A X. B) ~~ x <-> E.yE.z((y ~~ A /\ z ~~ B) /\ (y X. z) ~~ x))
33	14, 15, 32	3bitr4ri 269	. . . 4 |- ((A X. B) ~~ x <-> E.y e. Nc AE.z e. Nc Bx ~~ (y X. z))
34	33	abbii 2465	. . 3 |- {x | (A X. B) ~~ x} = {x | E.y e. Nc AE.z e. Nc Bx ~~ (y X. z)}
35	7, 8, 34	3eqtrri 2378	. 2 |- {x | E.y e. Nc AE.z e. Nc Bx ~~ (y X. z)} = Nc (A X. B)
36	6, 35	eqtri 2373	1 |- ( Nc A ·c Nc B) = Nc (A X. B)

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2615  _Vcvv 2859   class class class wbr 4639   X. cxp 4770  (class class class)co 5525  [cec 5945   ~~ 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:  muccl  6132  muccom  6134  mucass  6135  muc0  6142  mucid1  6143  addcdi  6250  muc0or  6252