Theorem cnfldmulg 13610

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 5885	. . . 4 |- ( x = 0 -> ( x (.g ` ℂfld ) B ) = ( 0 (.g ` ℂfld ) B ) )
2		oveq1 5885	. . . 4 |- ( x = 0 -> ( x x. B ) = ( 0 x. B ) )
3	1, 2	eqeq12d 2192	. . 3 |- ( x = 0 -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) ) )
4		oveq1 5885	. . . 4 |- ( x = y -> ( x (.g ` ℂfld ) B ) = ( y (.g ` ℂfld ) B ) )
5		oveq1 5885	. . . 4 |- ( x = y -> ( x x. B ) = ( y x. B ) )
6	4, 5	eqeq12d 2192	. . 3 |- ( x = y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( y (.g ` ℂfld ) B ) = ( y x. B ) ) )
7		oveq1 5885	. . . 4 |- ( x = ( y + 1 ) -> ( x (.g ` ℂfld ) B ) = ( ( y + 1 ) (.g ` ℂfld ) B ) )
8		oveq1 5885	. . . 4 |- ( x = ( y + 1 ) -> ( x x. B ) = ( ( y + 1 ) x. B ) )
9	7, 8	eqeq12d 2192	. . 3 |- ( x = ( y + 1 ) -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) )
10		oveq1 5885	. . . 4 |- ( x = -u y -> ( x (.g ` ℂfld ) B ) = ( -u y (.g ` ℂfld ) B ) )
11		oveq1 5885	. . . 4 |- ( x = -u y -> ( x x. B ) = ( -u y x. B ) )
12	10, 11	eqeq12d 2192	. . 3 |- ( x = -u y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) )
13		oveq1 5885	. . . 4 |- ( x = A -> ( x (.g ` ℂfld ) B ) = ( A (.g ` ℂfld ) B ) )
14		oveq1 5885	. . . 4 |- ( x = A -> ( x x. B ) = ( A x. B ) )
15	13, 14	eqeq12d 2192	. . 3 |- ( x = A -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( A (.g ` ℂfld ) B ) = ( A x. B ) ) )
16		cnfldbas 13599	. . . . 5 |- CC = ( Base ` ℂfld )
17		cnfld0 13605	. . . . 5 |- 0 = ( 0g ` ℂfld )
18		eqid 2177	. . . . 5 |- (.g ` ℂfld ) = (.g ` ℂfld )
19	16, 17, 18	mulg0 12994	. . . 4 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = 0 )
20		mul02 8347	. . . 4 |- ( B e. CC -> ( 0 x. B ) = 0 )
21	19, 20	eqtr4d 2213	. . 3 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) )
22		oveq1 5885	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y (.g ` ℂfld ) B ) + B ) = ( ( y x. B ) + B ) )
23		cnring 13604	. . . . . . . 8 |- ℂfld e. Ring
24		ringmnd 13195	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
25	23, 24	ax-mp 5	. . . . . . 7 |- ℂfld e. Mnd
26		cnfldadd 13600	. . . . . . . 8 |- + = ( +g ` ℂfld )
27	16, 18, 26	mulgnn0p1 13000	. . . . . . 7 |- ( (ℂfld e. Mnd /\ y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
28	25, 27	mp3an1 1324	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
29		nn0cn 9189	. . . . . . . 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 7987	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) x. B ) = ( ( y x. B ) + B ) )
33	28, 32	eqeq12d 2192	. . . . 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 5517	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
37		eqid 2177	. . . . . . 7 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
38	16, 18, 37	mulgnegnn 12999	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y (.g ` ℂfld ) B ) = ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) )
39		nncn 8930	. . . . . . . 8 |- ( y e. NN -> y e. CC )
40		mulneg1 8355	. . . . . . . 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 7941	. . . . . . . . 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 13607	. . . . . . . 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 2213	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y x. B ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
47	38, 46	eqeq12d 2192	. . . . 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 9374	. 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 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5878   CCcc 7812   0cc0 7814   1c1 7815   + caddc 7817   x. cmul 7819   -ucneg 8132   NNcn 8922   NN0cn0 9179   ZZcz 9256   Mndcmnd 12823   invgcminusg 12884  ._gcmg 12989   Ringcrg 13185  ℂ_fldccnfld 13595

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-addf 7936  ax-mulf 7937

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-tp 3602  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-5 8984  df-6 8985  df-7 8986  df-8 8987  df-9 8988  df-n0 9180  df-z 9257  df-dec 9388  df-uz 9532  df-fz 10012  df-seqfrec 10449  df-cj 10854  df-struct 12467  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-plusg 12552  df-mulr 12553  df-starv 12554  df-0g 12713  df-mgm 12781  df-sgrp 12814  df-mnd 12824  df-grp 12886  df-minusg 12887  df-mulg 12990  df-cmn 13096  df-mgp 13137  df-ring 13187  df-cring 13188  df-icnfld 13596

This theorem is referenced by:  zsssubrg  13619  zringmulg  13628