Theorem cnfldexp 14653

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 6035	. . . . 5 |- ( x = 0 -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) )
2		oveq2 6036	. . . . 5 |- ( x = 0 -> ( A ^ x ) = ( A ^ 0 ) )
3	1, 2	eqeq12d 2246	. . . 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 6035	. . . . 5 |- ( x = y -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( y (.g ` (mulGrp ` ℂfld ) ) A ) )
6		oveq2 6036	. . . . 5 |- ( x = y -> ( A ^ x ) = ( A ^ y ) )
7	5, 6	eqeq12d 2246	. . . 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 6035	. . . . 5 |- ( x = ( y + 1 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) )
10		oveq2 6036	. . . . 5 |- ( x = ( y + 1 ) -> ( A ^ x ) = ( A ^ ( y + 1 ) ) )
11	9, 10	eqeq12d 2246	. . . 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 6035	. . . . 5 |- ( x = B -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( B (.g ` (mulGrp ` ℂfld ) ) A ) )
14		oveq2 6036	. . . . 5 |- ( x = B -> ( A ^ x ) = ( A ^ B ) )
15	13, 14	eqeq12d 2246	. . . 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 14635	. . . . . 6 |- ℂfld e. _V
18		eqid 2231	. . . . . . 7 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
19		cnfldbas 14636	. . . . . . 7 |- CC = ( Base ` ℂfld )
20	18, 19	mgpbasg 14001	. . . . . 6 |- (ℂfld e. _V -> CC = ( Base ` (mulGrp ` ℂfld ) ) )
21	17, 20	ax-mp 5	. . . . 5 |- CC = ( Base ` (mulGrp ` ℂfld ) )
22		cnfld1 14648	. . . . . . 7 |- 1 = ( 1r ` ℂfld )
23	18, 22	ringidvalg 14036	. . . . . 6 |- (ℂfld e. _V -> 1 = ( 0g ` (mulGrp ` ℂfld ) ) )
24	17, 23	ax-mp 5	. . . . 5 |- 1 = ( 0g ` (mulGrp ` ℂfld ) )
25		eqid 2231	. . . . 5 |- (.g ` (mulGrp ` ℂfld ) ) = (.g ` (mulGrp ` ℂfld ) )
26	21, 24, 25	mulg0 13773	. . . 4 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = 1 )
27		exp0 10849	. . . 4 |- ( A e. CC -> ( A ^ 0 ) = 1 )
28	26, 27	eqtr4d 2267	. . 3 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ 0 ) )
29		oveq1 6035	. . . . . 6 |- ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) -> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) = ( ( A ^ y ) x. A ) )
30		cnring 14646	. . . . . . . . . 10 |- ℂfld e. Ring
31	18	ringmgp 14077	. . . . . . . . . 10 |- (ℂfld e. Ring -> (mulGrp ` ℂfld ) e. Mnd )
32	30, 31	ax-mp 5	. . . . . . . . 9 |- (mulGrp ` ℂfld ) e. Mnd
33		cnfldmul 14640	. . . . . . . . . . . 12 |- x. = ( .r ` ℂfld )
34	18, 33	mgpplusgg 13999	. . . . . . . . . . 11 |- (ℂfld e. _V -> x. = ( +g ` (mulGrp ` ℂfld ) ) )
35	17, 34	ax-mp 5	. . . . . . . . . 10 |- x. = ( +g ` (mulGrp ` ℂfld ) )
36	21, 25, 35	mulgnn0p1 13781	. . . . . . . . 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 10852	. . . . . . 7 |- ( ( A e. CC /\ y e. NN0 ) -> ( A ^ ( y + 1 ) ) = ( ( A ^ y ) x. A ) )
40	38, 39	eqeq12d 2246	. . . . . 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 9637	. 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 2202   _Vcvv 2803   ` cfv 5333  (class class class)co 6028   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   NN0cn0 9445   ^cexp 10844   Basecbs 13143   +g cplusg 13221   0gc0g 13400   Mndcmnd 13560  ._gcmg 13767  mulGrpcmgp 13995   Ringcrg 14071  ℂ_fldccnfld 14632

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-addf 8197  ax-mulf 8198

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-rp 9932  df-fz 10287  df-seqfrec 10754  df-exp 10845  df-cj 11463  df-abs 11620  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13149  df-sets 13150  df-plusg 13234  df-mulr 13235  df-starv 13236  df-tset 13240  df-ple 13241  df-ds 13243  df-unif 13244  df-0g 13402  df-topgen 13404  df-mgm 13500  df-sgrp 13546  df-mnd 13561  df-grp 13647  df-minusg 13648  df-mulg 13768  df-cmn 13934  df-mgp 13996  df-ur 14035  df-ring 14073  df-cring 14074  df-bl 14622  df-mopn 14623  df-fg 14625  df-metu 14626  df-cnfld 14633

This theorem is referenced by:  lgseisenlem4  15869