Theorem cnfldmulg 14064

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 5925	. . . 4 |- ( x = 0 -> ( x (.g ` ℂfld ) B ) = ( 0 (.g ` ℂfld ) B ) )
2		oveq1 5925	. . . 4 |- ( x = 0 -> ( x x. B ) = ( 0 x. B ) )
3	1, 2	eqeq12d 2208	. . 3 |- ( x = 0 -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) ) )
4		oveq1 5925	. . . 4 |- ( x = y -> ( x (.g ` ℂfld ) B ) = ( y (.g ` ℂfld ) B ) )
5		oveq1 5925	. . . 4 |- ( x = y -> ( x x. B ) = ( y x. B ) )
6	4, 5	eqeq12d 2208	. . 3 |- ( x = y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( y (.g ` ℂfld ) B ) = ( y x. B ) ) )
7		oveq1 5925	. . . 4 |- ( x = ( y + 1 ) -> ( x (.g ` ℂfld ) B ) = ( ( y + 1 ) (.g ` ℂfld ) B ) )
8		oveq1 5925	. . . 4 |- ( x = ( y + 1 ) -> ( x x. B ) = ( ( y + 1 ) x. B ) )
9	7, 8	eqeq12d 2208	. . 3 |- ( x = ( y + 1 ) -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) )
10		oveq1 5925	. . . 4 |- ( x = -u y -> ( x (.g ` ℂfld ) B ) = ( -u y (.g ` ℂfld ) B ) )
11		oveq1 5925	. . . 4 |- ( x = -u y -> ( x x. B ) = ( -u y x. B ) )
12	10, 11	eqeq12d 2208	. . 3 |- ( x = -u y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) )
13		oveq1 5925	. . . 4 |- ( x = A -> ( x (.g ` ℂfld ) B ) = ( A (.g ` ℂfld ) B ) )
14		oveq1 5925	. . . 4 |- ( x = A -> ( x x. B ) = ( A x. B ) )
15	13, 14	eqeq12d 2208	. . 3 |- ( x = A -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( A (.g ` ℂfld ) B ) = ( A x. B ) ) )
16		cnfldbas 14051	. . . . 5 |- CC = ( Base ` ℂfld )
17		cnfld0 14059	. . . . 5 |- 0 = ( 0g ` ℂfld )
18		eqid 2193	. . . . 5 |- (.g ` ℂfld ) = (.g ` ℂfld )
19	16, 17, 18	mulg0 13195	. . . 4 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = 0 )
20		mul02 8406	. . . 4 |- ( B e. CC -> ( 0 x. B ) = 0 )
21	19, 20	eqtr4d 2229	. . 3 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) )
22		oveq1 5925	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y (.g ` ℂfld ) B ) + B ) = ( ( y x. B ) + B ) )
23		cnring 14058	. . . . . . . 8 |- ℂfld e. Ring
24		ringmnd 13502	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
25	23, 24	ax-mp 5	. . . . . . 7 |- ℂfld e. Mnd
26		cnfldadd 14052	. . . . . . . 8 |- + = ( +g ` ℂfld )
27	16, 18, 26	mulgnn0p1 13203	. . . . . . 7 |- ( (ℂfld e. Mnd /\ y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
28	25, 27	mp3an1 1335	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
29		nn0cn 9250	. . . . . . . 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 8046	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) x. B ) = ( ( y x. B ) + B ) )
33	28, 32	eqeq12d 2208	. . . . 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 5554	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
37		eqid 2193	. . . . . . 7 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
38	16, 18, 37	mulgnegnn 13202	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y (.g ` ℂfld ) B ) = ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) )
39		nncn 8990	. . . . . . . 8 |- ( y e. NN -> y e. CC )
40		mulneg1 8414	. . . . . . . 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 7999	. . . . . . . . 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 14061	. . . . . . . 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 2229	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y x. B ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
47	38, 46	eqeq12d 2208	. . . . 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 9435	. 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 1364   e. wcel 2164   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   -ucneg 8191   NNcn 8982   NN0cn0 9240   ZZcz 9317   Mndcmnd 12997   invgcminusg 13073  ._gcmg 13189   Ringcrg 13492  ℂ_fldccnfld 14047

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-addf 7994  ax-mulf 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-z 9318  df-dec 9449  df-uz 9593  df-fz 10075  df-seqfrec 10519  df-cj 10986  df-struct 12620  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-starv 12710  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075  df-minusg 13076  df-mulg 13190  df-cmn 13356  df-mgp 13417  df-ring 13494  df-cring 13495  df-icnfld 14048

This theorem is referenced by:  zsssubrg  14073  zringmulg  14086  mulgrhm2  14098