Theorem expcl2lemap 10336

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 9092	. . 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 10335	. . . . . 6 |- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F )
6	5	ex 114	. . . . 5 |- ( A e. F -> ( B e. NN0 -> ( A ^ B ) e. F ) )
7	6	adantr 274	. . . 4 |- ( ( A e. F /\ A # 0 ) -> ( B e. NN0 -> ( A ^ B ) e. F ) )
8		simpll 519	. . . . . . . 8 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. F )
9	2, 8	sseldi 3100	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. CC )
10		simplr 520	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A # 0 )
11		simprl 521	. . . . . . . 8 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. RR )
12	11	recnd 7818	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. CC )
13		nnnn0 9008	. . . . . . . 8 |- ( -u B e. NN -> -u B e. NN0 )
14	13	ad2antll 483	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> -u B e. NN0 )
15		expineg2 10333	. . . . . . 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 1218	. . . . . 6 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) )
17		ssrab2 3187	. . . . . . . 8 |- { z e. F | z # 0 } C_ F
18		simpl 108	. . . . . . . . . 10 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A e. F /\ A # 0 ) )
19		breq1 3940	. . . . . . . . . . 11 |- ( z = A -> ( z # 0 <-> A # 0 ) )
20	19	elrab 2844	. . . . . . . . . 10 |- ( A e. { z e. F | z # 0 } <-> ( A e. F /\ A # 0 ) )
21	18, 20	sylibr 133	. . . . . . . . 9 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. { z e. F | z # 0 } )
22	17, 2	sstri 3111	. . . . . . . . . 10 |- { z e. F | z # 0 } C_ CC
23	17	sseli 3098	. . . . . . . . . . . 12 |- ( x e. { z e. F | z # 0 } -> x e. F )
24	17	sseli 3098	. . . . . . . . . . . 12 |- ( y e. { z e. F | z # 0 } -> y e. F )
25	23, 24, 3	syl2an 287	. . . . . . . . . . 11 |- ( ( x e. { z e. F | z # 0 } /\ y e. { z e. F | z # 0 } ) -> ( x x. y ) e. F )
26		breq1 3940	. . . . . . . . . . . . . 14 |- ( z = x -> ( z # 0 <-> x # 0 ) )
27	26	elrab 2844	. . . . . . . . . . . . 13 |- ( x e. { z e. F | z # 0 } <-> ( x e. F /\ x # 0 ) )
28	2	sseli 3098	. . . . . . . . . . . . . 14 |- ( x e. F -> x e. CC )
29	28	anim1i 338	. . . . . . . . . . . . 13 |- ( ( x e. F /\ x # 0 ) -> ( x e. CC /\ x # 0 ) )
30	27, 29	sylbi 120	. . . . . . . . . . . 12 |- ( x e. { z e. F | z # 0 } -> ( x e. CC /\ x # 0 ) )
31		breq1 3940	. . . . . . . . . . . . . 14 |- ( z = y -> ( z # 0 <-> y # 0 ) )
32	31	elrab 2844	. . . . . . . . . . . . 13 |- ( y e. { z e. F | z # 0 } <-> ( y e. F /\ y # 0 ) )
33	2	sseli 3098	. . . . . . . . . . . . . 14 |- ( y e. F -> y e. CC )
34	33	anim1i 338	. . . . . . . . . . . . 13 |- ( ( y e. F /\ y # 0 ) -> ( y e. CC /\ y # 0 ) )
35	32, 34	sylbi 120	. . . . . . . . . . . 12 |- ( y e. { z e. F | z # 0 } -> ( y e. CC /\ y # 0 ) )
36		mulap0 8439	. . . . . . . . . . . 12 |- ( ( ( x e. CC /\ x # 0 ) /\ ( y e. CC /\ y # 0 ) ) -> ( x x. y ) # 0 )
37	30, 35, 36	syl2an 287	. . . . . . . . . . 11 |- ( ( x e. { z e. F | z # 0 } /\ y e. { z e. F | z # 0 } ) -> ( x x. y ) # 0 )
38		breq1 3940	. . . . . . . . . . . 12 |- ( z = ( x x. y ) -> ( z # 0 <-> ( x x. y ) # 0 ) )
39	38	elrab 2844	. . . . . . . . . . 11 |- ( ( x x. y ) e. { z e. F | z # 0 } <-> ( ( x x. y ) e. F /\ ( x x. y ) # 0 ) )
40	25, 37, 39	sylanbrc 414	. . . . . . . . . 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 8376	. . . . . . . . . . 11 |- 1 # 0
42		breq1 3940	. . . . . . . . . . . 12 |- ( z = 1 -> ( z # 0 <-> 1 # 0 ) )
43	42	elrab 2844	. . . . . . . . . . 11 |- ( 1 e. { z e. F | z # 0 } <-> ( 1 e. F /\ 1 # 0 ) )
44	4, 41, 43	mpbir2an 927	. . . . . . . . . 10 |- 1 e. { z e. F | z # 0 }
45	22, 40, 44	expcllem 10335	. . . . . . . . 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 409	. . . . . . . 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	sseldi 3100	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. F )
48		breq1 3940	. . . . . . . . . 10 |- ( z = ( A ^ -u B ) -> ( z # 0 <-> ( A ^ -u B ) # 0 ) )
49	48	elrab 2844	. . . . . . . . 9 |- ( ( A ^ -u B ) e. { z e. F | z # 0 } <-> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) # 0 ) )
50	46, 49	sylib 121	. . . . . . . 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 113	. . . . . . 7 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) # 0 )
52		breq1 3940	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( x # 0 <-> ( A ^ -u B ) # 0 ) )
53		oveq2 5790	. . . . . . . . . 10 |- ( x = ( A ^ -u B ) -> ( 1 / x ) = ( 1 / ( A ^ -u B ) ) )
54	53	eleq1d 2209	. . . . . . . . 9 |- ( x = ( A ^ -u B ) -> ( ( 1 / x ) e. F <-> ( 1 / ( A ^ -u B ) ) e. F ) )
55	52, 54	imbi12d 233	. . . . . . . 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 114	. . . . . . . 8 |- ( x e. F -> ( x # 0 -> ( 1 / x ) e. F ) )
58	55, 57	vtoclga 2755	. . . . . . 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 2217	. . . . 5 |- ( ( ( A e. F /\ A # 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F )
61	60	ex 114	. . . 4 |- ( ( A e. F /\ A # 0 ) -> ( ( B e. RR /\ -u B e. NN ) -> ( A ^ B ) e. F ) )
62	7, 61	jaod 707	. . 3 |- ( ( A e. F /\ A # 0 ) -> ( ( B e. NN0 \/ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F ) )
63	1, 62	syl5bi 151	. 2 |- ( ( A e. F /\ A # 0 ) -> ( B e. ZZ -> ( A ^ B ) e. F ) )
64	63	3impia 1179	1 |- ( ( A e. F /\ A # 0 /\ B e. ZZ ) -> ( A ^ B ) e. F )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 963   = wceq 1332   e. wcel 1481   {crab 2421   C_ wss 3076   class class class wbr 3937  (class class class)co 5782   CCcc 7642   RRcr 7643   0cc0 7644   1c1 7645   x. cmul 7649   -ucneg 7958   # cap 8367   / cdiv 8456   NNcn 8744   NN0cn0 9001   ZZcz 9078   ^cexp 10323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-seqfrec 10250  df-exp 10324

This theorem is referenced by:  rpexpcl  10343  reexpclzap  10344  qexpclz  10345  m1expcl2  10346  expclzaplem  10348  1exp  10353