Theorem ghmmulg 13793

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 13790	. . . . . 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 13700	. . . . . 6 |- ( ( F e. ( G MndHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
6	1, 5	syl3an1 1304	. . . . 5 |- ( ( F e. ( G GrpHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
7	6	3expa 1227	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ N e. NN0 ) /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
8	7	an32s 568	. . 3 |- ( ( ( F e. ( G GrpHom H ) /\ X e. B ) /\ N e. NN0 ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
9	8	3adantl2 1178	. 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 1024	. . . . . . . 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 9376	. . . . . . . 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 1026	. . . . . . 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 13700	. . . . . . 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 1271	. . . . . 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 5631	. . . . 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 13782	. . . . . . . 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 9465	. . . . . . . 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 13676	. . . . . . 7 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
23	19, 21, 14, 22	syl3anc 1271	. . . . . 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 2229	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
25		eqid 2229	. . . . . . 7 |- ( invg ` H ) = ( invg ` H )
26	2, 24, 25	ghminv 13787	. . . . . 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 13783	. . . . . . 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 2229	. . . . . . . . 9 |- ( Base ` H ) = ( Base ` H )
31	2, 30	ghmf 13784	. . . . . . . 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 5771	. . . . . 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 13677	. . . . . 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 1271	. . . . 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 2272	. . . 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 13677	. . . . . . 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 1271	. . . . . 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 529	. . . . . . . . 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 8175	. . . . . . . 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 8448	. . . . . . 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 6016	. . . . . 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 2264	. . . . 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 5631	. . . 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 2264	. . 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 6016	. . 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 2264	. 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 1022	. . 3 |- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
49		elznn0nn 9460	. . 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 803	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 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   -->wf 5314   ` cfv 5318  (class class class)co 6001   RRcr 7998   -ucneg 8318   NNcn 9110   NN0cn0 9369   ZZcz 9446   Basecbs 13032   MndHom cmhm 13490   Grpcgrp 13533   invgcminusg 13534  ._gcmg 13656   GrpHom cghm 13777

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 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

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-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 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-map 6797  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-n0 9370  df-z 9447  df-uz 9723  df-seqfrec 10670  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-mhm 13492  df-grp 13536  df-minusg 13537  df-mulg 13657  df-ghm 13778

This theorem is referenced by:  mulgrhm2  14574