Theorem cnfldexp 14963

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 6092	. . . . 5 |- ( x = 0 -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) )
2		oveq2 6093	. . . . 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 6092	. . . . 5 |- ( x = y -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( y (.g ` (mulGrp ` ℂfld ) ) A ) )
6		oveq2 6093	. . . . 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 6092	. . . . 5 |- ( x = ( y + 1 ) -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( ( y + 1 ) (.g ` (mulGrp ` ℂfld ) ) A ) )
10		oveq2 6093	. . . . 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 6092	. . . . 5 |- ( x = B -> ( x (.g ` (mulGrp ` ℂfld ) ) A ) = ( B (.g ` (mulGrp ` ℂfld ) ) A ) )
14		oveq2 6093	. . . . 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 14945	. . . . . 6 |- ℂfld e. _V
18		eqid 2238	. . . . . . 7 |- (mulGrp ` ℂfld ) = (mulGrp ` ℂfld )
19		cnfldbas 14946	. . . . . . 7 |- CC = ( Base ` ℂfld )
20	18, 19	mgpbasg 14273	. . . . . 6 |- (ℂfld e. _V -> CC = ( Base ` (mulGrp ` ℂfld ) ) )
21	17, 20	ax-mp 5	. . . . 5 |- CC = ( Base ` (mulGrp ` ℂfld ) )
22		cnfld1 14958	. . . . . . 7 |- 1 = ( 1r ` ℂfld )
23	18, 22	ringidvalg 14313	. . . . . 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 13977	. . . 4 |- ( A e. CC -> ( 0 (.g ` (mulGrp ` ℂfld ) ) A ) = 1 )
27		exp0 10993	. . . 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 6092	. . . . . 6 |- ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) = ( A ^ y ) -> ( ( y (.g ` (mulGrp ` ℂfld ) ) A ) x. A ) = ( ( A ^ y ) x. A ) )
30		cnring 14956	. . . . . . . . . 10 |- ℂfld e. Ring
31	18	ringmgp 14355	. . . . . . . . . 10 |- (ℂfld e. Ring -> (mulGrp ` ℂfld ) e. Mnd )
32	30, 31	ax-mp 5	. . . . . . . . 9 |- (mulGrp ` ℂfld ) e. Mnd
33		cnfldmul 14950	. . . . . . . . . . . 12 |- x. = ( .r ` ℂfld )
34	18, 33	mgpplusgg 14270	. . . . . . . . . . 11 |- (ℂfld e. _V -> x. = ( +g ` (mulGrp ` ℂfld ) ) )
35	17, 34	ax-mp 5	. . . . . . . . . 10 |- x. = ( +g ` (mulGrp ` ℂfld ) )
36	21, 25, 35	mulgnn0p1 13985	. . . . . . . . 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 10996	. . . . . . 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 9764	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NN0cn0 9567   ^cexp 10988   Basecbs 13401   +g cplusg 13480   0gc0g 13659   Mndcmnd 13778  ._gcmg 13971  mulGrpcmgp 14266   Ringcrg 14349  ℂ_fldccnfld 14942

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-addf 8301  ax-mulf 8302

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-5 9368  df-6 9369  df-7 9370  df-8 9371  df-9 9372  df-n0 9568  df-z 9649  df-dec 9782  df-uz 9931  df-rp 10065  df-fz 10422  df-seqfrec 10898  df-exp 10989  df-cj 11621  df-abs 11779  df-struct 13403  df-ndx 13404  df-slot 13405  df-base 13407  df-sets 13408  df-plusg 13493  df-mulr 13494  df-starv 13495  df-tset 13499  df-ple 13500  df-ds 13502  df-unif 13503  df-0g 13661  df-topgen 13663  df-mgm 13725  df-sgrp 13766  df-mnd 13779  df-grp 13857  df-minusg 13858  df-mulg 13972  df-cmn 14138  df-mgp 14267  df-ur 14312  df-ring 14351  df-cring 14352  df-bl 14932  df-mopn 14933  df-fg 14935  df-metu 14936  df-cnfld 14943

This theorem is used by:  lgseisenlem4  16290