Theorem expcl2lem 12872

Description: Lemma for proving integer exponentiation closure laws. (Contributed by Mario Carneiro, 4-Jun-2014.) (Revised by Mario Carneiro, 9-Sep-2014.)

Hypotheses

Ref	Expression
expcllem.1	|- F C_ CC
expcllem.2	|- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )
expcllem.3	|- 1 e. F
expcl2lem.4	|- ( ( x e. F /\ x =/= 0 ) -> ( 1 / x ) e. F )

Assertion

Ref	Expression
expcl2lem	|- ( ( A e. F /\ A =/= 0 /\ B e. ZZ ) -> ( A ^ B ) e. F )

Distinct variable groups:   x, y, A   x, B   x, F, y

Allowed substitution hint:   B( y)

Proof of Theorem expcl2lem

Step	Hyp	Ref	Expression
1		elznn0nn 11391	. . 3 |- ( B e. ZZ <-> ( B e. NN0 \/ ( B e. RR /\ -u B e. NN ) ) )
2		expcllem.1	. . . . . . 7 |- F C_ CC
3		expcllem.2	. . . . . . 7 |- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F )
4		expcllem.3	. . . . . . 7 |- 1 e. F
5	2, 3, 4	expcllem 12871	. . . . . 6 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )
6	5	ex 450	. . . . 5 |- ( A e. F -> ( B e. NN0 -> ( A ^ B ) e. F ) )
7	6	adantr 481	. . . 4 |- ( ( A e. F /\ A =/= 0 ) -> ( B e. NN0 -> ( A ^ B ) e. F ) )
8		simpll 790	. . . . . . . 8 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. F )
9	2, 8	sseldi 3601	. . . . . . 7 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. CC )
10		simprl 794	. . . . . . . 8 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. RR )
11	10	recnd 10068	. . . . . . 7 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. CC )
12		nnnn0 11299	. . . . . . . 8 |- ( -u B e. NN -> -u B e. NN0 )
13	12	ad2antll 765	. . . . . . 7 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> -u B e. NN0 )
14		expneg2 12869	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ -u B e. NN0 ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) )
15	9, 11, 13, 14	syl3anc 1326	. . . . . 6 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) )
16		difss 3737	. . . . . . . 8 |- ( F \ { 0 } ) C_ F
17		simpl 473	. . . . . . . . . 10 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A e. F /\ A =/= 0 ) )
18		eldifsn 4317	. . . . . . . . . 10 |- ( A e. ( F \ { 0 } ) <-> ( A e. F /\ A =/= 0 ) )
19	17, 18	sylibr 224	. . . . . . . . 9 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. ( F \ { 0 } ) )
20	16, 2	sstri 3612	. . . . . . . . . 10 |- ( F \ { 0 } ) C_ CC
21	16	sseli 3599	. . . . . . . . . . . 12 |- ( x e. ( F \ { 0 } ) -> x e. F )
22	16	sseli 3599	. . . . . . . . . . . 12 |- ( y e. ( F \ { 0 } ) -> y e. F )
23	21, 22, 3	syl2an 494	. . . . . . . . . . 11 |- ( ( x e. ( F \ { 0 } ) /\ y e. ( F \ { 0 } ) ) -> ( x x. y ) e. F )
24		eldifsn 4317	. . . . . . . . . . . . 13 |- ( x e. ( F \ { 0 } ) <-> ( x e. F /\ x =/= 0 ) )
25	2	sseli 3599	. . . . . . . . . . . . . 14 |- ( x e. F -> x e. CC )
26	25	anim1i 592	. . . . . . . . . . . . 13 |- ( ( x e. F /\ x =/= 0 ) -> ( x e. CC /\ x =/= 0 ) )
27	24, 26	sylbi 207	. . . . . . . . . . . 12 |- ( x e. ( F \ { 0 } ) -> ( x e. CC /\ x =/= 0 ) )
28		eldifsn 4317	. . . . . . . . . . . . 13 |- ( y e. ( F \ { 0 } ) <-> ( y e. F /\ y =/= 0 ) )
29	2	sseli 3599	. . . . . . . . . . . . . 14 |- ( y e. F -> y e. CC )
30	29	anim1i 592	. . . . . . . . . . . . 13 |- ( ( y e. F /\ y =/= 0 ) -> ( y e. CC /\ y =/= 0 ) )
31	28, 30	sylbi 207	. . . . . . . . . . . 12 |- ( y e. ( F \ { 0 } ) -> ( y e. CC /\ y =/= 0 ) )
32		mulne0 10669	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 )
33	27, 31, 32	syl2an 494	. . . . . . . . . . 11 |- ( ( x e. ( F \ { 0 } ) /\ y e. ( F \ { 0 } ) ) -> ( x x. y ) =/= 0 )
34		eldifsn 4317	. . . . . . . . . . 11 |- ( ( x x. y ) e. ( F \ { 0 } ) <-> ( ( x x. y ) e. F /\ ( x x. y ) =/= 0 ) )
35	23, 33, 34	sylanbrc 698	. . . . . . . . . 10 |- ( ( x e. ( F \ { 0 } ) /\ y e. ( F \ { 0 } ) ) -> ( x x. y ) e. ( F \ { 0 } ) )
36		ax-1ne0 10005	. . . . . . . . . . 11 |- 1 =/= 0
37		eldifsn 4317	. . . . . . . . . . 11 |- ( 1 e. ( F \ { 0 } ) <-> ( 1 e. F /\ 1 =/= 0 ) )
38	4, 36, 37	mpbir2an 955	. . . . . . . . . 10 |- 1 e. ( F \ { 0 } )
39	20, 35, 38	expcllem 12871	. . . . . . . . 9 |- ( ( A e. ( F \ { 0 } ) /\ -u B e. NN0 ) -> ( A ^ -u B ) e. ( F \ { 0 } ) )
40	19, 13, 39	syl2anc 693	. . . . . . . 8 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. ( F \ { 0 } ) )
41	16, 40	sseldi 3601	. . . . . . 7 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. F )
42		eldifsn 4317	. . . . . . . . 9 |- ( ( A ^ -u B ) e. ( F \ { 0 } ) <-> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) =/= 0 ) )
43	40, 42	sylib 208	. . . . . . . 8 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) =/= 0 ) )
44	43	simprd 479	. . . . . . 7 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) =/= 0 )
45		neeq1 2856	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( x =/= 0 <-> ( A ^ -u B ) =/= 0 ) )
46		oveq2 6658	. . . . . . . . . 10 |- ( x = ( A ^ -u B ) -> ( 1 / x ) = ( 1 / ( A ^ -u B ) ) )
47	46	eleq1d 2686	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( ( 1 / x ) e. F <-> ( 1 / ( A ^ -u B ) ) e. F ) )
48	45, 47	imbi12d 334	. . . . . . . 8 |- ( x = ( A ^ -u B ) -> ( ( x =/= 0 -> ( 1 / x ) e. F ) <-> ( ( A ^ -u B ) =/= 0 -> ( 1 / ( A ^ -u B ) ) e. F ) ) )
49		expcl2lem.4	. . . . . . . . 9 |- ( ( x e. F /\ x =/= 0 ) -> ( 1 / x ) e. F )
50	49	ex 450	. . . . . . . 8 |- ( x e. F -> ( x =/= 0 -> ( 1 / x ) e. F ) )
51	48, 50	vtoclga 3272	. . . . . . 7 |- ( ( A ^ -u B ) e. F -> ( ( A ^ -u B ) =/= 0 -> ( 1 / ( A ^ -u B ) ) e. F ) )
52	41, 44, 51	sylc 65	. . . . . 6 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( 1 / ( A ^ -u B ) ) e. F )
53	15, 52	eqeltrd 2701	. . . . 5 |- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F )
54	53	ex 450	. . . 4 |- ( ( A e. F /\ A =/= 0 ) -> ( ( B e. RR /\ -u B e. NN ) -> ( A ^ B ) e. F ) )
55	7, 54	jaod 395	. . 3 |- ( ( A e. F /\ A =/= 0 ) -> ( ( B e. NN0 \/ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F ) )
56	1, 55	syl5bi 232	. 2 |- ( ( A e. F /\ A =/= 0 ) -> ( B e. ZZ -> ( A ^ B ) e. F ) )
57	56	3impia 1261	1 |- ( ( A e. F /\ A =/= 0 /\ B e. ZZ ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   {csn 4177  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   -ucneg 10267   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  rpexpcl  12879  reexpclz  12880  qexpclz  12881  m1expcl2  12882  expclzlem  12884  1exp  12889