Theorem cnfldmulg 13696

Description: The group multiple function in the field of complex numbers. (Contributed by Mario Carneiro, 14-Jun-2015.)

Assertion

Ref	Expression
cnfldmulg	|- ( ( A e. ZZ /\ B e. CC ) -> ( A (.g ` ℂfld ) B ) = ( A x. B ) )

Proof of Theorem cnfldmulg

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

Step	Hyp	Ref	Expression
1		oveq1 5895	. . . 4 |- ( x = 0 -> ( x (.g ` ℂfld ) B ) = ( 0 (.g ` ℂfld ) B ) )
2		oveq1 5895	. . . 4 |- ( x = 0 -> ( x x. B ) = ( 0 x. B ) )
3	1, 2	eqeq12d 2202	. . 3 |- ( x = 0 -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) ) )
4		oveq1 5895	. . . 4 |- ( x = y -> ( x (.g ` ℂfld ) B ) = ( y (.g ` ℂfld ) B ) )
5		oveq1 5895	. . . 4 |- ( x = y -> ( x x. B ) = ( y x. B ) )
6	4, 5	eqeq12d 2202	. . 3 |- ( x = y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( y (.g ` ℂfld ) B ) = ( y x. B ) ) )
7		oveq1 5895	. . . 4 |- ( x = ( y + 1 ) -> ( x (.g ` ℂfld ) B ) = ( ( y + 1 ) (.g ` ℂfld ) B ) )
8		oveq1 5895	. . . 4 |- ( x = ( y + 1 ) -> ( x x. B ) = ( ( y + 1 ) x. B ) )
9	7, 8	eqeq12d 2202	. . 3 |- ( x = ( y + 1 ) -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) )
10		oveq1 5895	. . . 4 |- ( x = -u y -> ( x (.g ` ℂfld ) B ) = ( -u y (.g ` ℂfld ) B ) )
11		oveq1 5895	. . . 4 |- ( x = -u y -> ( x x. B ) = ( -u y x. B ) )
12	10, 11	eqeq12d 2202	. . 3 |- ( x = -u y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) )
13		oveq1 5895	. . . 4 |- ( x = A -> ( x (.g ` ℂfld ) B ) = ( A (.g ` ℂfld ) B ) )
14		oveq1 5895	. . . 4 |- ( x = A -> ( x x. B ) = ( A x. B ) )
15	13, 14	eqeq12d 2202	. . 3 |- ( x = A -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( A (.g ` ℂfld ) B ) = ( A x. B ) ) )
16		cnfldbas 13685	. . . . 5 |- CC = ( Base ` ℂfld )
17		cnfld0 13691	. . . . 5 |- 0 = ( 0g ` ℂfld )
18		eqid 2187	. . . . 5 |- (.g ` ℂfld ) = (.g ` ℂfld )
19	16, 17, 18	mulg0 13017	. . . 4 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = 0 )
20		mul02 8357	. . . 4 |- ( B e. CC -> ( 0 x. B ) = 0 )
21	19, 20	eqtr4d 2223	. . 3 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) )
22		oveq1 5895	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y (.g ` ℂfld ) B ) + B ) = ( ( y x. B ) + B ) )
23		cnring 13690	. . . . . . . 8 |- ℂfld e. Ring
24		ringmnd 13243	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
25	23, 24	ax-mp 5	. . . . . . 7 |- ℂfld e. Mnd
26		cnfldadd 13686	. . . . . . . 8 |- + = ( +g ` ℂfld )
27	16, 18, 26	mulgnn0p1 13023	. . . . . . 7 |- ( (ℂfld e. Mnd /\ y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
28	25, 27	mp3an1 1334	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
29		nn0cn 9199	. . . . . . . 8 |- ( y e. NN0 -> y e. CC )
30	29	adantr 276	. . . . . . 7 |- ( ( y e. NN0 /\ B e. CC ) -> y e. CC )
31		simpr 110	. . . . . . 7 |- ( ( y e. NN0 /\ B e. CC ) -> B e. CC )
32	30, 31	adddirp1d 7997	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) x. B ) = ( ( y x. B ) + B ) )
33	28, 32	eqeq12d 2202	. . . . 5 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) <-> ( ( y (.g ` ℂfld ) B ) + B ) = ( ( y x. B ) + B ) ) )
34	22, 33	imbitrrid 156	. . . 4 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) )
35	34	expcom 116	. . 3 |- ( B e. CC -> ( y e. NN0 -> ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) ) )
36		fveq2 5527	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
37		eqid 2187	. . . . . . 7 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
38	16, 18, 37	mulgnegnn 13022	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y (.g ` ℂfld ) B ) = ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) )
39		nncn 8940	. . . . . . . 8 |- ( y e. NN -> y e. CC )
40		mulneg1 8365	. . . . . . . 8 |- ( ( y e. CC /\ B e. CC ) -> ( -u y x. B ) = -u ( y x. B ) )
41	39, 40	sylan 283	. . . . . . 7 |- ( ( y e. NN /\ B e. CC ) -> ( -u y x. B ) = -u ( y x. B ) )
42		mulcl 7951	. . . . . . . . 9 |- ( ( y e. CC /\ B e. CC ) -> ( y x. B ) e. CC )
43	39, 42	sylan 283	. . . . . . . 8 |- ( ( y e. NN /\ B e. CC ) -> ( y x. B ) e. CC )
44		cnfldneg 13693	. . . . . . . 8 |- ( ( y x. B ) e. CC -> ( ( invg ` ℂfld ) ` ( y x. B ) ) = -u ( y x. B ) )
45	43, 44	syl 14	. . . . . . 7 |- ( ( y e. NN /\ B e. CC ) -> ( ( invg ` ℂfld ) ` ( y x. B ) ) = -u ( y x. B ) )
46	41, 45	eqtr4d 2223	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y x. B ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
47	38, 46	eqeq12d 2202	. . . . 5 |- ( ( y e. NN /\ B e. CC ) -> ( ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) <-> ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) ) )
48	36, 47	imbitrrid 156	. . . 4 |- ( ( y e. NN /\ B e. CC ) -> ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) )
49	48	expcom 116	. . 3 |- ( B e. CC -> ( y e. NN -> ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) ) )
50	3, 6, 9, 12, 15, 21, 35, 49	zindd 9384	. 2 |- ( B e. CC -> ( A e. ZZ -> ( A (.g ` ℂfld ) B ) = ( A x. B ) ) )
51	50	impcom 125	1 |- ( ( A e. ZZ /\ B e. CC ) -> ( A (.g ` ℂfld ) B ) = ( A x. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825   + caddc 7827   x. cmul 7829   -ucneg 8142   NNcn 8932   NN0cn0 9189   ZZcz 9266   Mndcmnd 12836   invgcminusg 12897  ._gcmg 13011   Ringcrg 13233  ℂ_fldccnfld 13681

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-addf 7946  ax-mulf 7947

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-tp 3612  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-7 8996  df-8 8997  df-9 8998  df-n0 9190  df-z 9267  df-dec 9398  df-uz 9542  df-fz 10022  df-seqfrec 10459  df-cj 10864  df-struct 12477  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-starv 12565  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-minusg 12900  df-mulg 13012  df-cmn 13120  df-mgp 13163  df-ring 13235  df-cring 13236  df-icnfld 13682

This theorem is referenced by:  zsssubrg  13705  zringmulg  13714