Theorem expcl2lemap 10545

Description: Lemma for proving integer exponentiation closure laws. (Contributed by Jim Kingdon, 8-Jun-2020.)

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
expcl2lemap.4	|- ( ( x e. F /\ x # 0 ) -> ( 1 / x ) e. F )

Assertion

Ref	Expression
expcl2lemap	|- ( ( 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 expcl2lemap

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elznn0nn 9280	. . 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 10544	. . . . . 6 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )
6	5	ex 115	. . . . 5 |- ( A e. F -> ( B e. NN0 -> ( A ^ B ) e. F ) )
7	6	adantr 276	. . . 4 |- ( ( A e. F /\ A # 0 ) -> ( B e. NN0 -> ( A ^ B ) e. F ) )
8		simpll 527	. . . . . . . 8 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. F )
9	2, 8	sselid 3165	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. CC )
10		simplr 528	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A # 0 )
11		simprl 529	. . . . . . . 8 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. RR )
12	11	recnd 7999	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. CC )
13		nnnn0 9196	. . . . . . . 8 |- ( -u B e. NN -> -u B e. NN0 )
14	13	ad2antll 491	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> -u B e. NN0 )
15		expineg2 10542	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( B e. CC /\ -u B e. NN0 ) ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) )
16	9, 10, 12, 14, 15	syl22anc 1249	. . . . . 6 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) )
17		ssrab2 3252	. . . . . . . 8 |- { z e. F | z # 0 } C_ F
18		simpl 109	. . . . . . . . . 10 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A e. F /\ A # 0 ) )
19		breq1 4018	. . . . . . . . . . 11 |- ( z = A -> ( z # 0 <-> A # 0 ) )
20	19	elrab 2905	. . . . . . . . . 10 |- ( A e. { z e. F | z # 0 } <-> ( A e. F /\ A # 0 ) )
21	18, 20	sylibr 134	. . . . . . . . 9 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. { z e. F | z # 0 } )
22	17, 2	sstri 3176	. . . . . . . . . 10 |- { z e. F | z # 0 } C_ CC
23	17	sseli 3163	. . . . . . . . . . . 12 |- ( x e. { z e. F | z # 0 } -> x e. F )
24	17	sseli 3163	. . . . . . . . . . . 12 |- ( y e. { z e. F | z # 0 } -> y e. F )
25	23, 24, 3	syl2an 289	. . . . . . . . . . 11 |- ( ( x e. { z e. F | z # 0 } /\ y e. { z e. F | z # 0 } ) -> ( x x. y ) e. F )
26		breq1 4018	. . . . . . . . . . . . . 14 |- ( z = x -> ( z # 0 <-> x # 0 ) )
27	26	elrab 2905	. . . . . . . . . . . . 13 |- ( x e. { z e. F | z # 0 } <-> ( x e. F /\ x # 0 ) )
28	2	sseli 3163	. . . . . . . . . . . . . 14 |- ( x e. F -> x e. CC )
29	28	anim1i 340	. . . . . . . . . . . . 13 |- ( ( x e. F /\ x # 0 ) -> ( x e. CC /\ x # 0 ) )
30	27, 29	sylbi 121	. . . . . . . . . . . 12 |- ( x e. { z e. F | z # 0 } -> ( x e. CC /\ x # 0 ) )
31		breq1 4018	. . . . . . . . . . . . . 14 |- ( z = y -> ( z # 0 <-> y # 0 ) )
32	31	elrab 2905	. . . . . . . . . . . . 13 |- ( y e. { z e. F | z # 0 } <-> ( y e. F /\ y # 0 ) )
33	2	sseli 3163	. . . . . . . . . . . . . 14 |- ( y e. F -> y e. CC )
34	33	anim1i 340	. . . . . . . . . . . . 13 |- ( ( y e. F /\ y # 0 ) -> ( y e. CC /\ y # 0 ) )
35	32, 34	sylbi 121	. . . . . . . . . . . 12 |- ( y e. { z e. F | z # 0 } -> ( y e. CC /\ y # 0 ) )
36		mulap0 8624	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ x # 0 ) /\ ( y e. CC /\ y # 0 ) ) -> ( x x. y ) # 0 )
37	30, 35, 36	syl2an 289	. . . . . . . . . . 11 |- ( ( x e. { z e. F | z # 0 } /\ y e. { z e. F | z # 0 } ) -> ( x x. y ) # 0 )
38		breq1 4018	. . . . . . . . . . . 12 |- ( z = ( x x. y ) -> ( z # 0 <-> ( x x. y ) # 0 ) )
39	38	elrab 2905	. . . . . . . . . . 11 |- ( ( x x. y ) e. { z e. F | z # 0 } <-> ( ( x x. y ) e. F /\ ( x x. y ) # 0 ) )
40	25, 37, 39	sylanbrc 417	. . . . . . . . . 10 |- ( ( x e. { z e. F | z # 0 } /\ y e. { z e. F | z # 0 } ) -> ( x x. y ) e. { z e. F | z # 0 } )
41		1ap0 8560	. . . . . . . . . . 11 |- 1 # 0
42		breq1 4018	. . . . . . . . . . . 12 |- ( z = 1 -> ( z # 0 <-> 1 # 0 ) )
43	42	elrab 2905	. . . . . . . . . . 11 |- ( 1 e. { z e. F | z # 0 } <-> ( 1 e. F /\ 1 # 0 ) )
44	4, 41, 43	mpbir2an 943	. . . . . . . . . 10 |- 1 e. { z e. F | z # 0 }
45	22, 40, 44	expcllem 10544	. . . . . . . . 9 |- ( ( A e. { z e. F | z # 0 } /\ -u B e. NN0 ) -> ( A ^ -u B ) e. { z e. F | z # 0 } )
46	21, 14, 45	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. { z e. F | z # 0 } )
47	17, 46	sselid 3165	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. F )
48		breq1 4018	. . . . . . . . . 10 |- ( z = ( A ^ -u B ) -> ( z # 0 <-> ( A ^ -u B ) # 0 ) )
49	48	elrab 2905	. . . . . . . . 9 |- ( ( A ^ -u B ) e. { z e. F | z # 0 } <-> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) # 0 ) )
50	46, 49	sylib 122	. . . . . . . 8 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) # 0 ) )
51	50	simprd 114	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) # 0 )
52		breq1 4018	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( x # 0 <-> ( A ^ -u B ) # 0 ) )
53		oveq2 5896	. . . . . . . . . 10 |- ( x = ( A ^ -u B ) -> ( 1 / x ) = ( 1 / ( A ^ -u B ) ) )
54	53	eleq1d 2256	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( ( 1 / x ) e. F <-> ( 1 / ( A ^ -u B ) ) e. F ) )
55	52, 54	imbi12d 234	. . . . . . . 8 |- ( x = ( A ^ -u B ) -> ( ( x # 0 -> ( 1 / x ) e. F ) <-> ( ( A ^ -u B ) # 0 -> ( 1 / ( A ^ -u B ) ) e. F ) ) )
56		expcl2lemap.4	. . . . . . . . 9 |- ( ( x e. F /\ x # 0 ) -> ( 1 / x ) e. F )
57	56	ex 115	. . . . . . . 8 |- ( x e. F -> ( x # 0 -> ( 1 / x ) e. F ) )
58	55, 57	vtoclga 2815	. . . . . . 7 |- ( ( A ^ -u B ) e. F -> ( ( A ^ -u B ) # 0 -> ( 1 / ( A ^ -u B ) ) e. F ) )
59	47, 51, 58	sylc 62	. . . . . 6 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( 1 / ( A ^ -u B ) ) e. F )
60	16, 59	eqeltrd 2264	. . . . 5 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F )
61	60	ex 115	. . . 4 |- ( ( A e. F /\ A # 0 ) -> ( ( B e. RR /\ -u B e. NN ) -> ( A ^ B ) e. F ) )
62	7, 61	jaod 718	. . 3 |- ( ( A e. F /\ A # 0 ) -> ( ( B e. NN0 \/ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F ) )
63	1, 62	biimtrid 152	. 2 |- ( ( A e. F /\ A # 0 ) -> ( B e. ZZ -> ( A ^ B ) e. F ) )
64	63	3impia 1201	1 |- ( ( A e. F /\ A # 0 /\ B e. ZZ ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 979   = wceq 1363   e. wcel 2158   {crab 2469   C_ wss 3141   class class class wbr 4015  (class class class)co 5888   CCcc 7822   RRcr 7823   0cc0 7824   1c1 7825   x. cmul 7829   -ucneg 8142   # cap 8551   / cdiv 8642   NNcn 8932   NN0cn0 9189   ZZcz 9266   ^cexp 10532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-seqfrec 10459  df-exp 10533

This theorem is referenced by:  rpexpcl  10552  reexpclzap  10553  qexpclz  10554  m1expcl2  10555  expclzaplem  10557  1exp  10562