Theorem cenc 6182

Description: Cardinal exponentiation in terms of cardinality. Theorem XI.2.39 of [Rosser] p. 382. (Contributed by SF, 6-Mar-2015.)

Hypotheses

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

Assertion

Ref	Expression
cenc	|- ( Nc ~P1 A ↑c Nc ~P1 B) = Nc (A ^m B)

Proof of Theorem cenc

Dummy variables a b g p t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnc 6126	. . . . . . . . 9 |- (~P1 p e. Nc ~P1 A <-> ~P1 p ~~ ~P1 A)
2		enpw1 6063	. . . . . . . . 9 |- (p ~~ A <-> ~P1 p ~~ ~P1 A)
3	1, 2	bitr4i 243	. . . . . . . 8 |- (~P1 p e. Nc ~P1 A <-> p ~~ A)
4		elnc 6126	. . . . . . . . 9 |- (~P1 t e. Nc ~P1 B <-> ~P1 t ~~ ~P1 B)
5		enpw1 6063	. . . . . . . . 9 |- (t ~~ B <-> ~P1 t ~~ ~P1 B)
6	4, 5	bitr4i 243	. . . . . . . 8 |- (~P1 t e. Nc ~P1 B <-> t ~~ B)
7		enmap1 6075	. . . . . . . . 9 |- (p ~~ A -> (p ^m t) ~~ (A ^m t))
8		enmap2 6069	. . . . . . . . 9 |- (t ~~ B -> (A ^m t) ~~ (A ^m B))
9		entr 6039	. . . . . . . . 9 |- (((p ^m t) ~~ (A ^m t) /\ (A ^m t) ~~ (A ^m B)) -> (p ^m t) ~~ (A ^m B))
10	7, 8, 9	syl2an 463	. . . . . . . 8 |- ((p ~~ A /\ t ~~ B) -> (p ^m t) ~~ (A ^m B))
11	3, 6, 10	syl2anb 465	. . . . . . 7 |- ((~P1 p e. Nc ~P1 A /\ ~P1 t e. Nc ~P1 B) -> (p ^m t) ~~ (A ^m B))
12		entr 6039	. . . . . . . 8 |- ((g ~~ (p ^m t) /\ (p ^m t) ~~ (A ^m B)) -> g ~~ (A ^m B))
13	12	ancoms 439	. . . . . . 7 |- (((p ^m t) ~~ (A ^m B) /\ g ~~ (p ^m t)) -> g ~~ (A ^m B))
14	11, 13	sylan 457	. . . . . 6 |- (((~P1 p e. Nc ~P1 A /\ ~P1 t e. Nc ~P1 B) /\ g ~~ (p ^m t)) -> g ~~ (A ^m B))
15	14	3impa 1146	. . . . 5 |- ((~P1 p e. Nc ~P1 A /\ ~P1 t e. Nc ~P1 B /\ g ~~ (p ^m t)) -> g ~~ (A ^m B))
16	15	exlimivv 1635	. . . 4 |- (E.pE.t(~P1 p e. Nc ~P1 A /\ ~P1 t e. Nc ~P1 B /\ g ~~ (p ^m t)) -> g ~~ (A ^m B))
17		cenc.1	. . . . . . 7 |- A e. _V
18	17	pw1ex 4304	. . . . . 6 |- ~P1 A e. _V
19	18	ncelncsi 6122	. . . . 5 |- Nc ~P1 A e. NC
20		cenc.2	. . . . . . 7 |- B e. _V
21	20	pw1ex 4304	. . . . . 6 |- ~P1 B e. _V
22	21	ncelncsi 6122	. . . . 5 |- Nc ~P1 B e. NC
23		elce 6176	. . . . 5 |- (( Nc ~P1 A e. NC /\ Nc ~P1 B e. NC ) -> (g e. ( Nc ~P1 A ↑c Nc ~P1 B) <-> E.pE.t(~P1 p e. Nc ~P1 A /\ ~P1 t e. Nc ~P1 B /\ g ~~ (p ^m t))))
24	19, 22, 23	mp2an 653	. . . 4 |- (g e. ( Nc ~P1 A ↑c Nc ~P1 B) <-> E.pE.t(~P1 p e. Nc ~P1 A /\ ~P1 t e. Nc ~P1 B /\ g ~~ (p ^m t)))
25		elnc 6126	. . . 4 |- (g e. Nc (A ^m B) <-> g ~~ (A ^m B))
26	16, 24, 25	3imtr4i 257	. . 3 |- (g e. ( Nc ~P1 A ↑c Nc ~P1 B) -> g e. Nc (A ^m B))
27	26	ssriv 3278	. 2 |- ( Nc ~P1 A ↑c Nc ~P1 B) C_ Nc (A ^m B)
28	18	ncid 6124	. . . . 5 |- ~P1 A e. Nc ~P1 A
29	21	ncid 6124	. . . . 5 |- ~P1 B e. Nc ~P1 B
30		pw1eq 4144	. . . . . . . . 9 |- (a = A -> ~P1 a = ~P1 A)
31	30	eleq1d 2419	. . . . . . . 8 |- (a = A -> (~P1 a e. Nc ~P1 A <-> ~P1 A e. Nc ~P1 A))
32	31	adantr 451	. . . . . . 7 |- ((a = A /\ b = B) -> (~P1 a e. Nc ~P1 A <-> ~P1 A e. Nc ~P1 A))
33		pw1eq 4144	. . . . . . . . 9 |- (b = B -> ~P1 b = ~P1 B)
34	33	eleq1d 2419	. . . . . . . 8 |- (b = B -> (~P1 b e. Nc ~P1 B <-> ~P1 B e. Nc ~P1 B))
35	34	adantl 452	. . . . . . 7 |- ((a = A /\ b = B) -> (~P1 b e. Nc ~P1 B <-> ~P1 B e. Nc ~P1 B))
36		oveq12 5533	. . . . . . . 8 |- ((a = A /\ b = B) -> (a ^m b) = (A ^m B))
37	36	breq2d 4652	. . . . . . 7 |- ((a = A /\ b = B) -> (g ~~ (a ^m b) <-> g ~~ (A ^m B)))
38	32, 35, 37	3anbi123d 1252	. . . . . 6 |- ((a = A /\ b = B) -> ((~P1 a e. Nc ~P1 A /\ ~P1 b e. Nc ~P1 B /\ g ~~ (a ^m b)) <-> (~P1 A e. Nc ~P1 A /\ ~P1 B e. Nc ~P1 B /\ g ~~ (A ^m B))))
39	17, 20, 38	spc2ev 2948	. . . . 5 |- ((~P1 A e. Nc ~P1 A /\ ~P1 B e. Nc ~P1 B /\ g ~~ (A ^m B)) -> E.aE.b(~P1 a e. Nc ~P1 A /\ ~P1 b e. Nc ~P1 B /\ g ~~ (a ^m b)))
40	28, 29, 39	mp3an12 1267	. . . 4 |- (g ~~ (A ^m B) -> E.aE.b(~P1 a e. Nc ~P1 A /\ ~P1 b e. Nc ~P1 B /\ g ~~ (a ^m b)))
41		elce 6176	. . . . 5 |- (( Nc ~P1 A e. NC /\ Nc ~P1 B e. NC ) -> (g e. ( Nc ~P1 A ↑c Nc ~P1 B) <-> E.aE.b(~P1 a e. Nc ~P1 A /\ ~P1 b e. Nc ~P1 B /\ g ~~ (a ^m b))))
42	19, 22, 41	mp2an 653	. . . 4 |- (g e. ( Nc ~P1 A ↑c Nc ~P1 B) <-> E.aE.b(~P1 a e. Nc ~P1 A /\ ~P1 b e. Nc ~P1 B /\ g ~~ (a ^m b)))
43	40, 25, 42	3imtr4i 257	. . 3 |- (g e. Nc (A ^m B) -> g e. ( Nc ~P1 A ↑c Nc ~P1 B))
44	43	ssriv 3278	. 2 |- Nc (A ^m B) C_ ( Nc ~P1 A ↑c Nc ~P1 B)
45	27, 44	eqssi 3289	1 |- ( Nc ~P1 A ↑c Nc ~P1 B) = Nc (A ^m B)

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  ~P₁ cpw1 4136   class class class wbr 4640  (class class class)co 5526   ^m cmap 6000   ~~ cen 6029   NC cncs 6089   Nc cnc 6092   ↑_c cce 6097

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-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-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-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-ce 6107

This theorem is referenced by:  ce0nnulb  6183  ceclb  6184  ce0  6191  ce2  6193