Theorem cnfldexp 14851

Description: The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)

Assertion

Ref	Expression
cnfldexp	|- ( ( A e. CC /\ B e. NN0 ) -> ( B (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ B ) )

Proof of Theorem cnfldexp

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6065	. . . . 5 |- ( x = 0 -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) )
2		oveq2 6066	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
3	1, 2	eqeq12d 2249	. . . 4 |- ( x = 0 -> ( ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) <-> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ 0 ) ) )
4	3	imbi2d 230	. . 3 |- ( x = 0 -> ( ( A e. CC -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) ) <-> ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ 0 ) ) ) )
5		oveq1 6065	. . . . 5 |- ( x = y -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( y (.g ` (mulGrp ` ℂfld ) ) A ) )
6		oveq2 6066	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
7	5, 6	eqeq12d 2249	. . . 4 |- ( x = y -> ( ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) <-> ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) ) )
8	7	imbi2d 230	. . 3 |- ( x = y -> ( ( A e. CC -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) ) <-> ( A e. CC -> ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) ) ) )
9		oveq1 6065	. . . . 5 |- ( x = ( y + 1 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) )
10		oveq2 6066	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
11	9, 10	eqeq12d 2249	. . . 4 |- ( x = ( y + 1 ) -> ( ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) <-> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( y + 1 ) ) ) )
12	11	imbi2d 230	. . 3 |- ( x = ( y + 1 ) -> ( ( A e. CC -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) ) <-> ( A e. CC -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( y + 1 ) ) ) ) )
13		oveq1 6065	. . . . 5 |- ( x = B -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( B (.g ` (mulGrp ` ℂfld ) ) A ) )
14		oveq2 6066	. . . . 5 |- ( x = B -> ( A ^ x ) = ( A ^ B ) )
15	13, 14	eqeq12d 2249	. . . 4 |- ( x = B -> ( ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) <-> ( B (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ B ) ) )
16	15	imbi2d 230	. . 3 |- ( x = B -> ( ( A e. CC -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ x ) ) <-> ( A e. CC -> ( B (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ B ) ) ) )
17		cnfldex 14833	. . . . . 6 |- ℂfld e. _V
18		eqid 2234	. . . . . . 7 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
19		cnfldbas 14834	. . . . . . 7 |- CC = ( Base ` ℂfld )
20	18, 19	mgpbasg 14165	. . . . . 6 |- (ℂfld e. _V -> CC = ( Base ` (mulGrp ` ℂfld ) ) )
21	17, 20	ax-mp 5	. . . . 5 |- CC = ( Base ` (mulGrp ` ℂfld ) )
22		cnfld1 14846	. . . . . . 7 |- 1 = ( 1r ` ℂfld )
23	18, 22	ringidvalg 14204	. . . . . 6 |- (ℂfld e. _V -> 1 = ( 0g ` (mulGrp ` ℂfld ) ) )
24	17, 23	ax-mp 5	. . . . 5 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
25		eqid 2234	. . . . 5 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
26	21, 24, 25	mulg0 13878	. . . 4 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = 1 )
27		exp0 10929	. . . 4 |- ( A e. CC -> ( A ^ 0 ) = 1 )
28	26, 27	eqtr4d 2270	. . 3 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ 0 ) )
29		oveq1 6065	. . . . . 6 |- ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) -> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) = ( ( A ^ y ) x. A ) )
30		cnring 14844	. . . . . . . . . 10 |- ℂfld e. Ring
31	18	ringmgp 14245	. . . . . . . . . 10 |- (ℂfld e. Ring -> (mulGrp ` ℂfld ) e. Mnd )
32	30, 31	ax-mp 5	. . . . . . . . 9 |- (mulGrp ` ℂfld ) e. Mnd
33		cnfldmul 14838	. . . . . . . . . . . 12 |- x. = ( .r ` ℂfld )
34	18, 33	mgpplusgg 14163	. . . . . . . . . . 11 |- (ℂfld e. _V -> x. = ( +g ` (mulGrp ` ℂfld ) ) )
35	17, 34	ax-mp 5	. . . . . . . . . 10 |- x. = ( +g ` (mulGrp ` ℂfld ) )
36	21, 25, 35	mulgnn0p1 13886	. . . . . . . . 9 |- ( ( (mulGrp ` ℂfld ) e. Mnd /\ y e. NN0 /\ A e. CC ) -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) )
37	32, 36	mp3an1 1361	. . . . . . . 8 |- ( ( y e. NN0 /\ A e. CC ) -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) )
38	37	ancoms 268	. . . . . . 7 |- ( ( A e. CC /\ y e. NN0 ) -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) )
39		expp1 10932	. . . . . . 7 |- ( ( A e. CC /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
40	38, 39	eqeq12d 2249	. . . . . 6 |- ( ( A e. CC /\ y e. NN0 ) -> ( ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( y + 1 ) ) <-> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) = ( ( A ^ y ) x. A ) ) )
41	29, 40	imbitrrid 156	. . . . 5 |- ( ( A e. CC /\ y e. NN0 ) -> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( y + 1 ) ) ) )
42	41	expcom 116	. . . 4 |- ( y e. NN0 -> ( A e. CC -> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( y + 1 ) ) ) ) )
43	42	a2d 26	. . 3 |- ( y e. NN0 -> ( ( A e. CC -> ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) ) -> ( A e. CC -> ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ ( y + 1 ) ) ) ) )
44	4, 8, 12, 16, 28, 43	nn0ind 9710	. 2 |- ( B e. NN0 -> ( A e. CC -> ( B (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ B ) ) )
45	44	impcom 125	1 |- ( ( A e. CC /\ B e. NN0 ) -> ( B (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   _Vcvv 2815   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   NN0cn0 9513   ^cexp 10924   Basecbs 13296   +g cplusg 13374   0gc0g 13553   Mndcmnd 13677  ._gcmg 13872  mulGrpcmgp 14159   Ringcrg 14239  ℂ_fldccnfld 14830

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-addf 8265  ax-mulf 8266

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-rp 10005  df-fz 10362  df-seqfrec 10834  df-exp 10925  df-cj 11552  df-abs 11709  df-struct 13298  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-starv 13389  df-tset 13393  df-ple 13394  df-ds 13396  df-unif 13397  df-0g 13555  df-topgen 13557  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-mulg 13873  df-cmn 14039  df-mgp 14160  df-ur 14203  df-ring 14241  df-cring 14242  df-bl 14820  df-mopn 14821  df-fg 14823  df-metu 14824  df-cnfld 14831

This theorem is referenced by:  lgseisenlem4  16072