Theorem ghmmulg 13923

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 13920	. . . . . 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 13830	. . . . . 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 9468	. . . . . . . 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 13830	. . . . . . 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 5652	. . . . 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 13912	. . . . . . . 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 9559	. . . . . . . 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 13806	. . . . . . 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 2231	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
25		eqid 2231	. . . . . . 7 |- ( invg ` H ) = ( invg ` H )
26	2, 24, 25	ghminv 13917	. . . . . 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 13913	. . . . . . 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 2231	. . . . . . . . 9 |- ( Base ` H ) = ( Base ` H )
31	2, 30	ghmf 13914	. . . . . . . 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 5791	. . . . . 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 13807	. . . . . 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 2274	. . . 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 13807	. . . . . . 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 8267	. . . . . . . 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 8540	. . . . . . 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 6043	. . . . . 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 2266	. . . . 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 5652	. . . 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 2266	. . 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 6043	. . 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 2266	. 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 9554	. . 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 2202   -->wf 5329   ` cfv 5333  (class class class)co 6028   RRcr 8091   -ucneg 8410   NNcn 9202   NN0cn0 9461   ZZcz 9540   Basecbs 13162   MndHom cmhm 13620   Grpcgrp 13663   invgcminusg 13664  ._gcmg 13786   GrpHom cghm 13907

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817  df-seqfrec 10773  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-mhm 13622  df-grp 13666  df-minusg 13667  df-mulg 13787  df-ghm 13908

This theorem is referenced by:  mulgrhm2  14706