Theorem expghmap 14579

Description: Exponentiation is a group homomorphism from addition to multiplication. (Contributed by Mario Carneiro, 18-Jun-2015.) (Revised by AV, 10-Jun-2019.) (Revised by Jim Kingdon, 11-Sep-2025.)

Hypotheses

Ref	Expression
expghm.m	|- M = (mulGrp ` ℂfld )
expghmap.u	|- U = ( M ↾s { z e. CC | z # 0 } )

Assertion

Ref	Expression
expghmap	|- ( ( A e. CC /\ A # 0 ) -> ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) )

Distinct variable group:   x, A, z

Allowed substitution hints:   U( x, z)   M( x, z)

Proof of Theorem expghmap

Dummy variables r s u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		expclzaplem 10793	. . . 4 |- ( ( A e. CC /\ A # 0 /\ x e. ZZ ) -> ( A ^ x ) e. { z e. CC | z # 0 } )
2	1	3expa 1227	. . 3 |- ( ( ( A e. CC /\ A # 0 ) /\ x e. ZZ ) -> ( A ^ x ) e. { z e. CC | z # 0 } )
3	2	fmpttd 5792	. 2 |- ( ( A e. CC /\ A # 0 ) -> ( x e. ZZ |-> ( A ^ x ) ) : ZZ --> { z e. CC | z # 0 } )
4		expaddzap 10813	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( A ^ ( u + v ) ) = ( ( A ^ u ) x. ( A ^ v ) ) )
5		eqid 2229	. . . . . 6 |- ( x e. ZZ |-> ( A ^ x ) ) = ( x e. ZZ |-> ( A ^ x ) )
6		oveq2 6015	. . . . . 6 |- ( x = ( u + v ) -> ( A ^ x ) = ( A ^ ( u + v ) ) )
7		zaddcl 9494	. . . . . . 7 |- ( ( u e. ZZ /\ v e. ZZ ) -> ( u + v ) e. ZZ )
8	7	adantl 277	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( u + v ) e. ZZ )
9		simpll 527	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> A e. CC )
10		simplr 528	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> A # 0 )
11	9, 10, 8	expclzapd 10908	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( A ^ ( u + v ) ) e. CC )
12	5, 6, 8, 11	fvmptd3 5730	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( A ^ ( u + v ) ) )
13		oveq2 6015	. . . . . . 7 |- ( x = u -> ( A ^ x ) = ( A ^ u ) )
14		simprl 529	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> u e. ZZ )
15	9, 10, 14	expclzapd 10908	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( A ^ u ) e. CC )
16	5, 13, 14, 15	fvmptd3 5730	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) = ( A ^ u ) )
17		oveq2 6015	. . . . . . 7 |- ( x = v -> ( A ^ x ) = ( A ^ v ) )
18		simprr 531	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> v e. ZZ )
19	9, 10, 18	expclzapd 10908	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( A ^ v ) e. CC )
20	5, 17, 18, 19	fvmptd3 5730	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) = ( A ^ v ) )
21	16, 20	oveq12d 6025	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) = ( ( A ^ u ) x. ( A ^ v ) ) )
22	4, 12, 21	3eqtr4d 2272	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ ( u e. ZZ /\ v e. ZZ ) ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) )
23	22	ralrimivva 2612	. . 3 |- ( ( A e. CC /\ A # 0 ) -> A. u e. ZZ A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) )
24		simplr 528	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> u e. ZZ )
25	15	anassrs 400	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( A ^ u ) e. CC )
26	5, 13, 24, 25	fvmptd3 5730	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) = ( A ^ u ) )
27	26, 25	eqeltrd 2306	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) e. CC )
28		simpr 110	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> v e. ZZ )
29	19	anassrs 400	. . . . . . . . 9 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( A ^ v ) e. CC )
30	5, 17, 28, 29	fvmptd3 5730	. . . . . . . 8 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) = ( A ^ v ) )
31	30, 29	eqeltrd 2306	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) e. CC )
32	27, 31	mulcld 8175	. . . . . . 7 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) e. CC )
33		oveq1 6014	. . . . . . . 8 |- ( r = ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) -> ( r x. s ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. s ) )
34		oveq2 6015	. . . . . . . 8 |- ( s = ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) -> ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. s ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) )
35		eqid 2229	. . . . . . . 8 |- ( r e. CC , s e. CC |-> ( r x. s ) ) = ( r e. CC , s e. CC |-> ( r x. s ) )
36	33, 34, 35	ovmpog 6145	. . . . . . 7 |- ( ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) e. CC /\ ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) e. CC /\ ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) e. CC ) -> ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) )
37	27, 31, 32, 36	syl3anc 1271	. . . . . 6 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) )
38	37	eqeq2d 2241	. . . . 5 |- ( ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) /\ v e. ZZ ) -> ( ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) <-> ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) ) )
39	38	ralbidva 2526	. . . 4 |- ( ( ( A e. CC /\ A # 0 ) /\ u e. ZZ ) -> ( A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) <-> A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) ) )
40	39	ralbidva 2526	. . 3 |- ( ( A e. CC /\ A # 0 ) -> ( A. u e. ZZ A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) <-> A. u e. ZZ A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) x. ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) ) )
41	23, 40	mpbird 167	. 2 |- ( ( A e. CC /\ A # 0 ) -> A. u e. ZZ A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) )
42		zringgrp 14567	. . . 4 |- ℤring e. Grp
43		cnring 14542	. . . . 5 |- ℂfld e. Ring
44		cnfldui 14561	. . . . . 6 |- { z e. CC | z # 0 } = (Unit ` ℂfld )
45		expghmap.u	. . . . . . 7 |- U = ( M ↾s { z e. CC | z # 0 } )
46		expghm.m	. . . . . . . 8 |- M = (mulGrp ` ℂfld )
47	46	oveq1i 6017	. . . . . . 7 |- ( M ↾s { z e. CC | z # 0 } ) = ( (mulGrp ` ℂfld ) ↾s { z e. CC | z # 0 } )
48	45, 47	eqtri 2250	. . . . . 6 |- U = ( (mulGrp ` ℂfld ) ↾s { z e. CC | z # 0 } )
49	44, 48	unitgrp 14088	. . . . 5 |- (ℂfld e. Ring -> U e. Grp )
50	43, 49	ax-mp 5	. . . 4 |- U e. Grp
51	42, 50	pm3.2i 272	. . 3 |- (ℤring e. Grp /\ U e. Grp )
52		zringbas 14568	. . . 4 |- ZZ = ( Base ` ℤring )
53	45	a1i 9	. . . . . 6 |- ( T. -> U = ( M ↾s { z e. CC | z # 0 } ) )
54		cnfldbas 14532	. . . . . . . 8 |- CC = ( Base ` ℂfld )
55	46, 54	mgpbasg 13897	. . . . . . 7 |- (ℂfld e. Ring -> CC = ( Base ` M ) )
56	43, 55	mp1i 10	. . . . . 6 |- ( T. -> CC = ( Base ` M ) )
57	46	mgpex 13896	. . . . . . 7 |- (ℂfld e. Ring -> M e. _V )
58	43, 57	mp1i 10	. . . . . 6 |- ( T. -> M e. _V )
59		apsscn 8802	. . . . . . 7 |- { z e. CC | z # 0 } C_ CC
60	59	a1i 9	. . . . . 6 |- ( T. -> { z e. CC | z # 0 } C_ CC )
61	53, 56, 58, 60	ressbas2d 13109	. . . . 5 |- ( T. -> { z e. CC | z # 0 } = ( Base ` U ) )
62	61	mptru 1404	. . . 4 |- { z e. CC | z # 0 } = ( Base ` U )
63		zringplusg 14569	. . . 4 |- + = ( +g ` ℤring )
64		mpocnfldmul 14535	. . . . . . . 8 |- ( r e. CC , s e. CC |-> ( r x. s ) ) = ( .r ` ℂfld )
65	46, 64	mgpplusgg 13895	. . . . . . 7 |- (ℂfld e. Ring -> ( r e. CC , s e. CC |-> ( r x. s ) ) = ( +g ` M ) )
66	43, 65	mp1i 10	. . . . . 6 |- ( T. -> ( r e. CC , s e. CC |-> ( r x. s ) ) = ( +g ` M ) )
67		cnex 8131	. . . . . . . 8 |- CC e. _V
68	67	rabex 4228	. . . . . . 7 |- { z e. CC | z # 0 } e. _V
69	68	a1i 9	. . . . . 6 |- ( T. -> { z e. CC | z # 0 } e. _V )
70	53, 66, 69, 58	ressplusgd 13170	. . . . 5 |- ( T. -> ( r e. CC , s e. CC |-> ( r x. s ) ) = ( +g ` U ) )
71	70	mptru 1404	. . . 4 |- ( r e. CC , s e. CC |-> ( r x. s ) ) = ( +g ` U )
72	52, 62, 63, 71	isghm 13788	. . 3 |- ( ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) <-> ( (ℤring e. Grp /\ U e. Grp ) /\ ( ( x e. ZZ |-> ( A ^ x ) ) : ZZ --> { z e. CC | z # 0 } /\ A. u e. ZZ A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) ) ) )
73	51, 72	mpbiran 946	. 2 |- ( ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) <-> ( ( x e. ZZ |-> ( A ^ x ) ) : ZZ --> { z e. CC | z # 0 } /\ A. u e. ZZ A. v e. ZZ ( ( x e. ZZ |-> ( A ^ x ) ) ` ( u + v ) ) = ( ( ( x e. ZZ |-> ( A ^ x ) ) ` u ) ( r e. CC , s e. CC |-> ( r x. s ) ) ( ( x e. ZZ |-> ( A ^ x ) ) ` v ) ) ) )
74	3, 41, 73	sylanbrc 417	1 |- ( ( A e. CC /\ A # 0 ) -> ( x e. ZZ |-> ( A ^ x ) ) e. (ℤring GrpHom U ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   T. wtru 1396   e. wcel 2200   A.wral 2508   {crab 2512   _Vcvv 2799   C_ wss 3197   class class class wbr 4083   |-> cmpt 4145   -->wf 5314   ` cfv 5318  (class class class)co 6007   e. cmpo 6009   CCcc 8005   0cc0 8007   + caddc 8010   x. cmul 8012   # cap 8736   ZZcz 9454   ^cexp 10768   Basecbs 13040   ↾_s cress 13041   +g cplusg 13118   Grpcgrp 13541   GrpHom cghm 13785  mulGrpcmgp 13891   Ringcrg 13967  ℂ_fldccnfld 14528  ℤ_ringczring 14562

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-addf 8129  ax-mulf 8130

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-tpos 6397  df-recs 6457  df-frec 6543  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-5 9180  df-6 9181  df-7 9182  df-8 9183  df-9 9184  df-n0 9378  df-z 9455  df-dec 9587  df-uz 9731  df-rp 9858  df-fz 10213  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-abs 11518  df-struct 13042  df-ndx 13043  df-slot 13044  df-base 13046  df-sets 13047  df-iress 13048  df-plusg 13131  df-mulr 13132  df-starv 13133  df-tset 13137  df-ple 13138  df-ds 13140  df-unif 13141  df-0g 13299  df-topgen 13301  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-minusg 13545  df-subg 13715  df-ghm 13786  df-cmn 13831  df-abl 13832  df-mgp 13892  df-ur 13931  df-srg 13935  df-ring 13969  df-cring 13970  df-oppr 14039  df-dvdsr 14060  df-unit 14061  df-subrg 14191  df-bl 14518  df-mopn 14519  df-fg 14521  df-metu 14522  df-cnfld 14529  df-zring 14563

This theorem is referenced by:  lgseisenlem4  15760