Theorem cnfldexp 14886

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 6082	. . . . 5 |- ( x = 0 -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) )
2		oveq2 6083	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
3	1, 2	eqeq12d 2253	. . . 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 6082	. . . . 5 |- ( x = y -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( y (.g ` (mulGrp ` ℂfld ) ) A ) )
6		oveq2 6083	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
7	5, 6	eqeq12d 2253	. . . 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 6082	. . . . 5 |- ( x = ( y + 1 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) )
10		oveq2 6083	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
11	9, 10	eqeq12d 2253	. . . 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 6082	. . . . 5 |- ( x = B -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( B (.g ` (mulGrp ` ℂfld ) ) A ) )
14		oveq2 6083	. . . . 5 |- ( x = B -> ( A ^ x ) = ( A ^ B ) )
15	13, 14	eqeq12d 2253	. . . 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 14868	. . . . . 6 |- ℂfld e. _V
18		eqid 2238	. . . . . . 7 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
19		cnfldbas 14869	. . . . . . 7 |- CC = ( Base ` ℂfld )
20	18, 19	mgpbasg 14200	. . . . . 6 |- (ℂfld e. _V -> CC = ( Base ` (mulGrp ` ℂfld ) ) )
21	17, 20	ax-mp 5	. . . . 5 |- CC = ( Base ` (mulGrp ` ℂfld ) )
22		cnfld1 14881	. . . . . . 7 |- 1 = ( 1r ` ℂfld )
23	18, 22	ringidvalg 14239	. . . . . 6 |- (ℂfld e. _V -> 1 = ( 0g ` (mulGrp ` ℂfld ) ) )
24	17, 23	ax-mp 5	. . . . 5 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
25		eqid 2238	. . . . 5 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
26	21, 24, 25	mulg0 13905	. . . 4 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = 1 )
27		exp0 10958	. . . 4 |- ( A e. CC -> ( A ^ 0 ) = 1 )
28	26, 27	eqtr4d 2274	. . 3 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ 0 ) )
29		oveq1 6082	. . . . . 6 |- ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) -> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) = ( ( A ^ y ) x. A ) )
30		cnring 14879	. . . . . . . . . 10 |- ℂfld e. Ring
31	18	ringmgp 14280	. . . . . . . . . 10 |- (ℂfld e. Ring -> (mulGrp ` ℂfld ) e. Mnd )
32	30, 31	ax-mp 5	. . . . . . . . 9 |- (mulGrp ` ℂfld ) e. Mnd
33		cnfldmul 14873	. . . . . . . . . . . 12 |- x. = ( .r ` ℂfld )
34	18, 33	mgpplusgg 14198	. . . . . . . . . . 11 |- (ℂfld e. _V -> x. = ( +g ` (mulGrp ` ℂfld ) ) )
35	17, 34	ax-mp 5	. . . . . . . . . 10 |- x. = ( +g ` (mulGrp ` ℂfld ) )
36	21, 25, 35	mulgnn0p1 13913	. . . . . . . . 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 1365	. . . . . . . 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 10961	. . . . . . 7 |- ( ( A e. CC /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
40	38, 39	eqeq12d 2253	. . . . . 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 9739	. 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 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5372  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NN0cn0 9542   ^cexp 10953   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706  ._gcmg 13899  mulGrpcmgp 14194   Ringcrg 14274  ℂ_fldccnfld 14865

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-addf 8291  ax-mulf 8292

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-rp 10034  df-fz 10391  df-seqfrec 10863  df-exp 10954  df-cj 11585  df-abs 11743  df-struct 13332  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-plusg 13421  df-mulr 13422  df-starv 13423  df-tset 13427  df-ple 13428  df-ds 13430  df-unif 13431  df-0g 13589  df-topgen 13591  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-grp 13785  df-minusg 13786  df-mulg 13900  df-cmn 14066  df-mgp 14195  df-ur 14238  df-ring 14276  df-cring 14277  df-bl 14855  df-mopn 14856  df-fg 14858  df-metu 14859  df-cnfld 14866

This theorem is referenced by:  lgseisenlem4  16106