Theorem ghmmulg 14009

Description: A group homomorphism preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
ghmmulg.b	|- B = ( Base ` G )
ghmmulg.s	|- .x. = (.g ` G )
ghmmulg.t	|- .X. = (.g ` H )

Assertion

Ref	Expression
ghmmulg	|- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )

Proof of Theorem ghmmulg

Step	Hyp	Ref	Expression
1		ghmmhm 14006	. . . . . 6 |- ( F e. ( G GrpHom H ) -> F e. ( G MndHom H ) )
2		ghmmulg.b	. . . . . . 7 |- B = ( Base ` G )
3		ghmmulg.s	. . . . . . 7 |- .x. = (.g ` G )
4		ghmmulg.t	. . . . . . 7 |- .X. = (.g ` H )
5	2, 3, 4	mhmmulg 13916	. . . . . 6 |- ( ( F e. ( G MndHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
6	1, 5	syl3an1 1307	. . . . 5 |- ( ( F e. ( G GrpHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
7	6	3expa 1230	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ N e. NN0 ) /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
8	7	an32s 570	. . 3 |- ( ( ( F e. ( G GrpHom H ) /\ X e. B ) /\ N e. NN0 ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
9	8	3adantl2 1181	. 2 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ N e. NN0 ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
10		simpl1 1027	. . . . . . . 8 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> F e. ( G GrpHom H ) )
11	10, 1	syl 14	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> F e. ( G MndHom H ) )
12		nnnn0 9520	. . . . . . . 8 |- ( -u N e. NN -> -u N e. NN0 )
13	12	ad2antll 491	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
14		simpl3 1029	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> X e. B )
15	2, 3, 4	mhmmulg 13916	. . . . . . 7 |- ( ( F e. ( G MndHom H ) /\ -u N e. NN0 /\ X e. B ) -> ( F ` ( -u N .x. X ) ) = ( -u N .X. ( F ` X ) ) )
16	11, 13, 14, 15	syl3anc 1274	. . . . . 6 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( F ` ( -u N .x. X ) ) = ( -u N .X. ( F ` X ) ) )
17	16	fveq2d 5679	. . . . 5 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( invg ` H ) ` ( F ` ( -u N .x. X ) ) ) = ( ( invg ` H ) ` ( -u N .X. ( F ` X ) ) ) )
18		ghmgrp1 13998	. . . . . . . 8 |- ( F e. ( G GrpHom H ) -> G e. Grp )
19	10, 18	syl 14	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> G e. Grp )
20		nnz 9613	. . . . . . . 8 |- ( -u N e. NN -> -u N e. ZZ )
21	20	ad2antll 491	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
22	2, 3	mulgcl 13892	. . . . . . 7 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
23	19, 21, 14, 22	syl3anc 1274	. . . . . 6 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u N .x. X ) e. B )
24		eqid 2234	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
25		eqid 2234	. . . . . . 7 |- ( invg ` H ) = ( invg ` H )
26	2, 24, 25	ghminv 14003	. . . . . 6 |- ( ( F e. ( G GrpHom H ) /\ ( -u N .x. X ) e. B ) -> ( F ` ( ( invg ` G ) ` ( -u N .x. X ) ) ) = ( ( invg ` H ) ` ( F ` ( -u N .x. X ) ) ) )
27	10, 23, 26	syl2anc 411	. . . . 5 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( F ` ( ( invg ` G ) ` ( -u N .x. X ) ) ) = ( ( invg ` H ) ` ( F ` ( -u N .x. X ) ) ) )
28		ghmgrp2 13999	. . . . . . 7 |- ( F e. ( G GrpHom H ) -> H e. Grp )
29	10, 28	syl 14	. . . . . 6 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> H e. Grp )
30		eqid 2234	. . . . . . . . 9 |- ( Base ` H ) = ( Base ` H )
31	2, 30	ghmf 14000	. . . . . . . 8 |- ( F e. ( G GrpHom H ) -> F : B --> ( Base ` H ) )
32	10, 31	syl 14	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> F : B --> ( Base ` H ) )
33	32, 14	ffvelcdmd 5818	. . . . . 6 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( F ` X ) e. ( Base ` H ) )
34	30, 4, 25	mulgneg 13893	. . . . . 6 |- ( ( H e. Grp /\ -u N e. ZZ /\ ( F ` X ) e. ( Base ` H ) ) -> ( -u -u N .X. ( F ` X ) ) = ( ( invg ` H ) ` ( -u N .X. ( F ` X ) ) ) )
35	29, 21, 33, 34	syl3anc 1274	. . . . 5 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .X. ( F ` X ) ) = ( ( invg ` H ) ` ( -u N .X. ( F ` X ) ) ) )
36	17, 27, 35	3eqtr4d 2277	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( F ` ( ( invg ` G ) ` ( -u N .x. X ) ) ) = ( -u -u N .X. ( F ` X ) ) )
37	2, 3, 24	mulgneg 13893	. . . . . . 7 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u -u N .x. X ) = ( ( invg ` G ) ` ( -u N .x. X ) ) )
38	19, 21, 14, 37	syl3anc 1274	. . . . . 6 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .x. X ) = ( ( invg ` G ) ` ( -u N .x. X ) ) )
39		simprl 531	. . . . . . . . 9 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
40	39	recnd 8318	. . . . . . . 8 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
41	40	negnegd 8591	. . . . . . 7 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u -u N = N )
42	41	oveq1d 6073	. . . . . 6 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .x. X ) = ( N .x. X ) )
43	38, 42	eqtr3d 2269	. . . . 5 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( invg ` G ) ` ( -u N .x. X ) ) = ( N .x. X ) )
44	43	fveq2d 5679	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( F ` ( ( invg ` G ) ` ( -u N .x. X ) ) ) = ( F ` ( N .x. X ) ) )
45	36, 44	eqtr3d 2269	. . 3 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .X. ( F ` X ) ) = ( F ` ( N .x. X ) ) )
46	41	oveq1d 6073	. . 3 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( -u -u N .X. ( F ` X ) ) = ( N .X. ( F ` X ) ) )
47	45, 46	eqtr3d 2269	. 2 |- ( ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
48		simp2 1025	. . 3 |- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
49		elznn0nn 9608	. . 3 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
50	48, 49	sylib 122	. 2 |- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
51	9, 47, 50	mpjaodan 806	1 |- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2205   -->wf 5353   ` cfv 5357  (class class class)co 6058   RRcr 8142   -ucneg 8461   NNcn 9254   NN0cn0 9513   ZZcz 9594   Basecbs 13296   MndHom cmhm 13712   Grpcgrp 13755   invgcminusg 13756  ._gcmg 13872   GrpHom cghm 13993

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834  df-ndx 13299  df-slot 13300  df-base 13302  df-plusg 13387  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-mhm 13714  df-grp 13758  df-minusg 13759  df-mulg 13873  df-ghm 13994

This theorem is referenced by:  mulgrhm2  14884