Theorem cnfldmulg 14556

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 6014	. . . 4 |- ( x = 0 -> ( x (.g ` ℂfld ) B ) = ( 0 (.g ` ℂfld ) B ) )
2		oveq1 6014	. . . 4 |- ( x = 0 -> ( x x. B ) = ( 0 x. B ) )
3	1, 2	eqeq12d 2244	. . 3 |- ( x = 0 -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) ) )
4		oveq1 6014	. . . 4 |- ( x = y -> ( x (.g ` ℂfld ) B ) = ( y (.g ` ℂfld ) B ) )
5		oveq1 6014	. . . 4 |- ( x = y -> ( x x. B ) = ( y x. B ) )
6	4, 5	eqeq12d 2244	. . 3 |- ( x = y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( y (.g ` ℂfld ) B ) = ( y x. B ) ) )
7		oveq1 6014	. . . 4 |- ( x = ( y + 1 ) -> ( x (.g ` ℂfld ) B ) = ( ( y + 1 ) (.g ` ℂfld ) B ) )
8		oveq1 6014	. . . 4 |- ( x = ( y + 1 ) -> ( x x. B ) = ( ( y + 1 ) x. B ) )
9	7, 8	eqeq12d 2244	. . 3 |- ( x = ( y + 1 ) -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) )
10		oveq1 6014	. . . 4 |- ( x = -u y -> ( x (.g ` ℂfld ) B ) = ( -u y (.g ` ℂfld ) B ) )
11		oveq1 6014	. . . 4 |- ( x = -u y -> ( x x. B ) = ( -u y x. B ) )
12	10, 11	eqeq12d 2244	. . 3 |- ( x = -u y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) )
13		oveq1 6014	. . . 4 |- ( x = A -> ( x (.g ` ℂfld ) B ) = ( A (.g ` ℂfld ) B ) )
14		oveq1 6014	. . . 4 |- ( x = A -> ( x x. B ) = ( A x. B ) )
15	13, 14	eqeq12d 2244	. . 3 |- ( x = A -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( A (.g ` ℂfld ) B ) = ( A x. B ) ) )
16		cnfldbas 14540	. . . . 5 |- CC = ( Base ` ℂfld )
17		cnfld0 14551	. . . . 5 |- 0 = ( 0g ` ℂfld )
18		eqid 2229	. . . . 5 |- (.g ` ℂfld ) = (.g ` ℂfld )
19	16, 17, 18	mulg0 13678	. . . 4 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = 0 )
20		mul02 8544	. . . 4 |- ( B e. CC -> ( 0 x. B ) = 0 )
21	19, 20	eqtr4d 2265	. . 3 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) )
22		oveq1 6014	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y (.g ` ℂfld ) B ) + B ) = ( ( y x. B ) + B ) )
23		cnring 14550	. . . . . . . 8 |- ℂfld e. Ring
24		ringmnd 13985	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
25	23, 24	ax-mp 5	. . . . . . 7 |- ℂfld e. Mnd
26		cnfldadd 14542	. . . . . . . 8 |- + = ( +g ` ℂfld )
27	16, 18, 26	mulgnn0p1 13686	. . . . . . 7 |- ( (ℂfld e. Mnd /\ y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
28	25, 27	mp3an1 1358	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
29		nn0cn 9390	. . . . . . . 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 8184	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) x. B ) = ( ( y x. B ) + B ) )
33	28, 32	eqeq12d 2244	. . . . 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 5629	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
37		eqid 2229	. . . . . . 7 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
38	16, 18, 37	mulgnegnn 13685	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y (.g ` ℂfld ) B ) = ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) )
39		nncn 9129	. . . . . . . 8 |- ( y e. NN -> y e. CC )
40		mulneg1 8552	. . . . . . . 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 8137	. . . . . . . . 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 14553	. . . . . . . 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 2265	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y x. B ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
47	38, 46	eqeq12d 2244	. . . . 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 9576	. 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 1395   e. wcel 2200   ` cfv 5318  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   + caddc 8013   x. cmul 8015   -ucneg 8329   NNcn 9121   NN0cn0 9380   ZZcz 9457   Mndcmnd 13465   invgcminusg 13550  ._gcmg 13672   Ringcrg 13975  ℂ_fldccnfld 14536

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-addf 8132  ax-mulf 8133

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-uz 9734  df-rp 9862  df-fz 10217  df-seqfrec 10682  df-cj 11369  df-abs 11526  df-struct 13050  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-plusg 13139  df-mulr 13140  df-starv 13141  df-tset 13145  df-ple 13146  df-ds 13148  df-unif 13149  df-0g 13307  df-topgen 13309  df-mgm 13405  df-sgrp 13451  df-mnd 13466  df-grp 13552  df-minusg 13553  df-mulg 13673  df-cmn 13839  df-mgp 13900  df-ring 13977  df-cring 13978  df-bl 14526  df-mopn 14527  df-fg 14529  df-metu 14530  df-cnfld 14537

This theorem is referenced by:  zsssubrg  14565  zringmulg  14578  mulgrhm2  14590