Theorem expcl2lemap 10713

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 9401	. . 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 10712	. . . . . 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 3195	. . . . . . 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 8116	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. CC )
13		nnnn0 9317	. . . . . . . 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 10710	. . . . . . 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 1251	. . . . . 6 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) )
17		ssrab2 3282	. . . . . . . 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 4053	. . . . . . . . . . 11 |- ( z = A -> ( z # 0 <-> A # 0 ) )
20	19	elrab 2933	. . . . . . . . . 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 3206	. . . . . . . . . 10 |- { z e. F | z # 0 } C_ CC
23	17	sseli 3193	. . . . . . . . . . . 12 |- ( x e. { z e. F | z # 0 } -> x e. F )
24	17	sseli 3193	. . . . . . . . . . . 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 4053	. . . . . . . . . . . . . 14 |- ( z = x -> ( z # 0 <-> x # 0 ) )
27	26	elrab 2933	. . . . . . . . . . . . 13 |- ( x e. { z e. F | z # 0 } <-> ( x e. F /\ x # 0 ) )
28	2	sseli 3193	. . . . . . . . . . . . . 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 4053	. . . . . . . . . . . . . 14 |- ( z = y -> ( z # 0 <-> y # 0 ) )
32	31	elrab 2933	. . . . . . . . . . . . 13 |- ( y e. { z e. F | z # 0 } <-> ( y e. F /\ y # 0 ) )
33	2	sseli 3193	. . . . . . . . . . . . . 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 8742	. . . . . . . . . . . 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 4053	. . . . . . . . . . . 12 |- ( z = ( x x. y ) -> ( z # 0 <-> ( x x. y ) # 0 ) )
39	38	elrab 2933	. . . . . . . . . . 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 8678	. . . . . . . . . . 11 |- 1 # 0
42		breq1 4053	. . . . . . . . . . . 12 |- ( z = 1 -> ( z # 0 <-> 1 # 0 ) )
43	42	elrab 2933	. . . . . . . . . . 11 |- ( 1 e. { z e. F | z # 0 } <-> ( 1 e. F /\ 1 # 0 ) )
44	4, 41, 43	mpbir2an 945	. . . . . . . . . 10 |- 1 e. { z e. F | z # 0 }
45	22, 40, 44	expcllem 10712	. . . . . . . . 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 3195	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. F )
48		breq1 4053	. . . . . . . . . 10 |- ( z = ( A ^ -u B ) -> ( z # 0 <-> ( A ^ -u B ) # 0 ) )
49	48	elrab 2933	. . . . . . . . 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 4053	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( x # 0 <-> ( A ^ -u B ) # 0 ) )
53		oveq2 5964	. . . . . . . . . 10 |- ( x = ( A ^ -u B ) -> ( 1 / x ) = ( 1 / ( A ^ -u B ) ) )
54	53	eleq1d 2275	. . . . . . . . 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 2841	. . . . . . 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 2283	. . . . 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 719	. . 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 1203	1 |- ( ( A e. F /\ A # 0 /\ B e. ZZ ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2177   {crab 2489   C_ wss 3170   class class class wbr 4050  (class class class)co 5956   CCcc 7938   RRcr 7939   0cc0 7940   1c1 7941   x. cmul 7945   -ucneg 8259   # cap 8669   / cdiv 8760   NNcn 9051   NN0cn0 9310   ZZcz 9387   ^cexp 10700

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057  ax-pre-mulext 8058

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-div 8761  df-inn 9052  df-n0 9311  df-z 9388  df-uz 9664  df-seqfrec 10610  df-exp 10701

This theorem is referenced by:  rpexpcl  10720  reexpclzap  10721  qexpclz  10722  m1expcl2  10723  expclzaplem  10725  1exp  10730