Theorem ghmmulg 14036

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 14033	. . . . . 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 13943	. . . . . 6 |- ( ( F e. ( G MndHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
6	1, 5	syl3an1 1311	. . . . 5 |- ( ( F e. ( G GrpHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
7	6	3expa 1234	. . . 4 |- ( ( ( F e. ( G GrpHom H ) /\ N e. NN0 ) /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
8	7	an32s 574	. . 3 |- ( ( ( F e. ( G GrpHom H ) /\ X e. B ) /\ N e. NN0 ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
9	8	3adantl2 1185	. 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 1031	. . . . . . . 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 9549	. . . . . . . 8 |- ( -u N e. NN -> -u N e. NN0 )
13	12	ad2antll 495	. . . . . . 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 1033	. . . . . . 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 13943	. . . . . . 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 1278	. . . . . 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 5694	. . . . 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 14025	. . . . . . . 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 9642	. . . . . . . 8 |- ( -u N e. NN -> -u N e. ZZ )
21	20	ad2antll 495	. . . . . . 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 13919	. . . . . . 7 |- ( ( G e. Grp /\ -u N e. ZZ /\ X e. B ) -> ( -u N .x. X ) e. B )
23	19, 21, 14, 22	syl3anc 1278	. . . . . 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 2238	. . . . . . 7 |- ( invg ` G ) = ( invg ` G )
25		eqid 2238	. . . . . . 7 |- ( invg ` H ) = ( invg ` H )
26	2, 24, 25	ghminv 14030	. . . . . 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 415	. . . . 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 14026	. . . . . . 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 2238	. . . . . . . . 9 |- ( Base ` H ) = ( Base ` H )
31	2, 30	ghmf 14027	. . . . . . . 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 5835	. . . . . 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 13920	. . . . . 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 1278	. . . . 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 2281	. . . 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 13920	. . . . . . 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 1278	. . . . . 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 535	. . . . . . . . 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 8344	. . . . . . . 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 8618	. . . . . . 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 6090	. . . . . 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 2273	. . . . 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 5694	. . . 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 2273	. . 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 6090	. . 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 2273	. 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 1029	. . 3 |- ( ( F e. ( G GrpHom H ) /\ N e. ZZ /\ X e. B ) -> N e. ZZ )
49		elznn0nn 9637	. . 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 810	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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   -->wf 5368   ` cfv 5372  (class class class)co 6075   RRcr 8168   -ucneg 8488   NNcn 9283   NN0cn0 9542   ZZcz 9623   Basecbs 13330   MndHom cmhm 13741   Grpcgrp 13782   invgcminusg 13783  ._gcmg 13899   GrpHom cghm 14020

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-grp 13785  df-minusg 13786  df-mulg 13900  df-ghm 14021

This theorem is referenced by:  mulgrhm2  14917