Theorem cnfldmulg 15022

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 6095	. . . 4 |- ( x = 0 -> ( x (.g ` ℂfld ) B ) = ( 0 (.g ` ℂfld ) B ) )
2		oveq1 6095	. . . 4 |- ( x = 0 -> ( x x. B ) = ( 0 x. B ) )
3	1, 2	eqeq12d 2253	. . 3 |- ( x = 0 -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) ) )
4		oveq1 6095	. . . 4 |- ( x = y -> ( x (.g ` ℂfld ) B ) = ( y (.g ` ℂfld ) B ) )
5		oveq1 6095	. . . 4 |- ( x = y -> ( x x. B ) = ( y x. B ) )
6	4, 5	eqeq12d 2253	. . 3 |- ( x = y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( y (.g ` ℂfld ) B ) = ( y x. B ) ) )
7		oveq1 6095	. . . 4 |- ( x = ( y + 1 ) -> ( x (.g ` ℂfld ) B ) = ( ( y + 1 ) (.g ` ℂfld ) B ) )
8		oveq1 6095	. . . 4 |- ( x = ( y + 1 ) -> ( x x. B ) = ( ( y + 1 ) x. B ) )
9	7, 8	eqeq12d 2253	. . 3 |- ( x = ( y + 1 ) -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y + 1 ) x. B ) ) )
10		oveq1 6095	. . . 4 |- ( x = -u y -> ( x (.g ` ℂfld ) B ) = ( -u y (.g ` ℂfld ) B ) )
11		oveq1 6095	. . . 4 |- ( x = -u y -> ( x x. B ) = ( -u y x. B ) )
12	10, 11	eqeq12d 2253	. . 3 |- ( x = -u y -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( -u y (.g ` ℂfld ) B ) = ( -u y x. B ) ) )
13		oveq1 6095	. . . 4 |- ( x = A -> ( x (.g ` ℂfld ) B ) = ( A (.g ` ℂfld ) B ) )
14		oveq1 6095	. . . 4 |- ( x = A -> ( x x. B ) = ( A x. B ) )
15	13, 14	eqeq12d 2253	. . 3 |- ( x = A -> ( ( x (.g ` ℂfld ) B ) = ( x x. B ) <-> ( A (.g ` ℂfld ) B ) = ( A x. B ) ) )
16		cnfldbas 15006	. . . . 5 |- CC = ( Base ` ℂfld )
17		cnfld0 15017	. . . . 5 |- 0 = ( 0g ` ℂfld )
18		eqid 2238	. . . . 5 |- (.g ` ℂfld ) = (.g ` ℂfld )
19	16, 17, 18	mulg0 13987	. . . 4 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = 0 )
20		mul02 8721	. . . 4 |- ( B e. CC -> ( 0 x. B ) = 0 )
21	19, 20	eqtr4d 2274	. . 3 |- ( B e. CC -> ( 0 (.g ` ℂfld ) B ) = ( 0 x. B ) )
22		oveq1 6095	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( y (.g ` ℂfld ) B ) + B ) = ( ( y x. B ) + B ) )
23		cnring 15016	. . . . . . . 8 |- ℂfld e. Ring
24		ringmnd 14417	. . . . . . . 8 |- (ℂfld e. Ring -> ℂfld e. Mnd )
25	23, 24	ax-mp 5	. . . . . . 7 |- ℂfld e. Mnd
26		cnfldadd 15008	. . . . . . . 8 |- + = ( +g ` ℂfld )
27	16, 18, 26	mulgnn0p1 13995	. . . . . . 7 |- ( (ℂfld e. Mnd /\ y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
28	25, 27	mp3an1 1365	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) (.g ` ℂfld ) B ) = ( ( y (.g ` ℂfld ) B ) + B ) )
29		nn0cn 9584	. . . . . . . 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 8358	. . . . . 6 |- ( ( y e. NN0 /\ B e. CC ) -> ( ( y + 1 ) x. B ) = ( ( y x. B ) + B ) )
33	28, 32	eqeq12d 2253	. . . . 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 5698	. . . . 5 |- ( ( y (.g ` ℂfld ) B ) = ( y x. B ) -> ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
37		eqid 2238	. . . . . . 7 |- ( invg ` ℂfld ) = ( invg ` ℂfld )
38	16, 18, 37	mulgnegnn 13994	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y (.g ` ℂfld ) B ) = ( ( invg ` ℂfld ) ` ( y (.g ` ℂfld ) B ) ) )
39		nncn 9321	. . . . . . . 8 |- ( y e. NN -> y e. CC )
40		mulneg1 8729	. . . . . . . 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 8312	. . . . . . . . 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 15019	. . . . . . . 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 2274	. . . . . 6 |- ( ( y e. NN /\ B e. CC ) -> ( -u y x. B ) = ( ( invg ` ℂfld ) ` ( y x. B ) ) )
47	38, 46	eqeq12d 2253	. . . . 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 9775	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   -ucneg 8505   NNcn 9313   NN0cn0 9574   ZZcz 9655   Mndcmnd 13788   invgcminusg 13865  ._gcmg 13981   Ringcrg 14407  ℂ_fldccnfld 15002

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-addf 8307  ax-mulf 8308

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-dec 9789  df-uz 9938  df-rp 10072  df-fz 10429  df-seqfrec 10906  df-cj 11629  df-abs 11787  df-struct 13412  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-starv 13505  df-tset 13509  df-ple 13510  df-ds 13512  df-unif 13513  df-0g 13671  df-topgen 13673  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-mulg 13982  df-cmn 14179  df-mgp 14325  df-ring 14409  df-cring 14410  df-bl 14992  df-mopn 14993  df-fg 14995  df-metu 14996  df-cnfld 15003

This theorem is used by:  zsssubrg  15031  zringmulg  15042  mulgrhm2  15054