Theorem mhmmulg 13943

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

Hypotheses

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

Assertion

Ref	Expression
mhmmulg	|- ( ( F e. ( G MndHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )

Proof of Theorem mhmmulg

Dummy variables m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvoveq1 6098	. . . . . 6 |- ( n = 0 -> ( F ` ( n .x. X ) ) = ( F ` ( 0 .x. X ) ) )
2		oveq1 6082	. . . . . 6 |- ( n = 0 -> ( n .X. ( F ` X ) ) = ( 0 .X. ( F ` X ) ) )
3	1, 2	eqeq12d 2253	. . . . 5 |- ( n = 0 -> ( ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) <-> ( F ` ( 0 .x. X ) ) = ( 0 .X. ( F ` X ) ) ) )
4	3	imbi2d 230	. . . 4 |- ( n = 0 -> ( ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) ) <-> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( 0 .x. X ) ) = ( 0 .X. ( F ` X ) ) ) ) )
5		fvoveq1 6098	. . . . . 6 |- ( n = m -> ( F ` ( n .x. X ) ) = ( F ` ( m .x. X ) ) )
6		oveq1 6082	. . . . . 6 |- ( n = m -> ( n .X. ( F ` X ) ) = ( m .X. ( F ` X ) ) )
7	5, 6	eqeq12d 2253	. . . . 5 |- ( n = m -> ( ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) <-> ( F ` ( m .x. X ) ) = ( m .X. ( F ` X ) ) ) )
8	7	imbi2d 230	. . . 4 |- ( n = m -> ( ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) ) <-> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( m .x. X ) ) = ( m .X. ( F ` X ) ) ) ) )
9		fvoveq1 6098	. . . . . 6 |- ( n = ( m + 1 ) -> ( F ` ( n .x. X ) ) = ( F ` ( ( m + 1 ) .x. X ) ) )
10		oveq1 6082	. . . . . 6 |- ( n = ( m + 1 ) -> ( n .X. ( F ` X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) )
11	9, 10	eqeq12d 2253	. . . . 5 |- ( n = ( m + 1 ) -> ( ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) <-> ( F ` ( ( m + 1 ) .x. X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) ) )
12	11	imbi2d 230	. . . 4 |- ( n = ( m + 1 ) -> ( ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) ) <-> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( ( m + 1 ) .x. X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) ) ) )
13		fvoveq1 6098	. . . . . 6 |- ( n = N -> ( F ` ( n .x. X ) ) = ( F ` ( N .x. X ) ) )
14		oveq1 6082	. . . . . 6 |- ( n = N -> ( n .X. ( F ` X ) ) = ( N .X. ( F ` X ) ) )
15	13, 14	eqeq12d 2253	. . . . 5 |- ( n = N -> ( ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) <-> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) ) )
16	15	imbi2d 230	. . . 4 |- ( n = N -> ( ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( n .x. X ) ) = ( n .X. ( F ` X ) ) ) <-> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) ) ) )
17		eqid 2238	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
18		eqid 2238	. . . . . . 7 |- ( 0g ` H ) = ( 0g ` H )
19	17, 18	mhm0 13752	. . . . . 6 |- ( F e. ( G MndHom H ) -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
20	19	adantr 276	. . . . 5 |- ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( 0g ` G ) ) = ( 0g ` H ) )
21		mhmmulg.b	. . . . . . . 8 |- B = ( Base ` G )
22		mhmmulg.s	. . . . . . . 8 |- .x. = (.g ` G )
23	21, 17, 22	mulg0 13905	. . . . . . 7 |- ( X e. B -> ( 0 .x. X ) = ( 0g ` G ) )
24	23	adantl 277	. . . . . 6 |- ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( 0 .x. X ) = ( 0g ` G ) )
25	24	fveq2d 5694	. . . . 5 |- ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( 0 .x. X ) ) = ( F ` ( 0g ` G ) ) )
26		eqid 2238	. . . . . . . 8 |- ( Base ` H ) = ( Base ` H )
27	21, 26	mhmf 13749	. . . . . . 7 |- ( F e. ( G MndHom H ) -> F : B --> ( Base ` H ) )
28	27	ffvelcdmda 5834	. . . . . 6 |- ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` X ) e. ( Base ` H ) )
29		mhmmulg.t	. . . . . . 7 |- .X. = (.g ` H )
30	26, 18, 29	mulg0 13905	. . . . . 6 |- ( ( F ` X ) e. ( Base ` H ) -> ( 0 .X. ( F ` X ) ) = ( 0g ` H ) )
31	28, 30	syl 14	. . . . 5 |- ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( 0 .X. ( F ` X ) ) = ( 0g ` H ) )
32	20, 25, 31	3eqtr4d 2281	. . . 4 |- ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( 0 .x. X ) ) = ( 0 .X. ( F ` X ) ) )
33		oveq1 6082	. . . . . . 7 |- ( ( F ` ( m .x. X ) ) = ( m .X. ( F ` X ) ) -> ( ( F ` ( m .x. X ) ) ( +g ` H ) ( F ` X ) ) = ( ( m .X. ( F ` X ) ) ( +g ` H ) ( F ` X ) ) )
34		mhmrcl1 13747	. . . . . . . . . . . 12 |- ( F e. ( G MndHom H ) -> G e. Mnd )
35	34	ad2antrr 492	. . . . . . . . . . 11 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> G e. Mnd )
36		simpr 110	. . . . . . . . . . 11 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> m e. NN0 )
37		simplr 533	. . . . . . . . . . 11 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> X e. B )
38		eqid 2238	. . . . . . . . . . . 12 |- ( +g ` G ) = ( +g ` G )
39	21, 22, 38	mulgnn0p1 13913	. . . . . . . . . . 11 |- ( ( G e. Mnd /\ m e. NN0 /\ X e. B ) -> ( ( m + 1 ) .x. X ) = ( ( m .x. X ) ( +g ` G ) X ) )
40	35, 36, 37, 39	syl3anc 1278	. . . . . . . . . 10 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( ( m + 1 ) .x. X ) = ( ( m .x. X ) ( +g ` G ) X ) )
41	40	fveq2d 5694	. . . . . . . . 9 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( F ` ( ( m + 1 ) .x. X ) ) = ( F ` ( ( m .x. X ) ( +g ` G ) X ) ) )
42		simpll 531	. . . . . . . . . 10 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> F e. ( G MndHom H ) )
43	34	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( F e. ( G MndHom H ) /\ m e. NN0 ) /\ X e. B ) -> G e. Mnd )
44		simplr 533	. . . . . . . . . . . 12 |- ( ( ( F e. ( G MndHom H ) /\ m e. NN0 ) /\ X e. B ) -> m e. NN0 )
45		simpr 110	. . . . . . . . . . . 12 |- ( ( ( F e. ( G MndHom H ) /\ m e. NN0 ) /\ X e. B ) -> X e. B )
46	21, 22	mulgnn0cl 13918	. . . . . . . . . . . 12 |- ( ( G e. Mnd /\ m e. NN0 /\ X e. B ) -> ( m .x. X ) e. B )
47	43, 44, 45, 46	syl3anc 1278	. . . . . . . . . . 11 |- ( ( ( F e. ( G MndHom H ) /\ m e. NN0 ) /\ X e. B ) -> ( m .x. X ) e. B )
48	47	an32s 574	. . . . . . . . . 10 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( m .x. X ) e. B )
49		eqid 2238	. . . . . . . . . . 11 |- ( +g ` H ) = ( +g ` H )
50	21, 38, 49	mhmlin 13751	. . . . . . . . . 10 |- ( ( F e. ( G MndHom H ) /\ ( m .x. X ) e. B /\ X e. B ) -> ( F ` ( ( m .x. X ) ( +g ` G ) X ) ) = ( ( F ` ( m .x. X ) ) ( +g ` H ) ( F ` X ) ) )
51	42, 48, 37, 50	syl3anc 1278	. . . . . . . . 9 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( F ` ( ( m .x. X ) ( +g ` G ) X ) ) = ( ( F ` ( m .x. X ) ) ( +g ` H ) ( F ` X ) ) )
52	41, 51	eqtrd 2271	. . . . . . . 8 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( F ` ( ( m + 1 ) .x. X ) ) = ( ( F ` ( m .x. X ) ) ( +g ` H ) ( F ` X ) ) )
53		mhmrcl2 13748	. . . . . . . . . 10 |- ( F e. ( G MndHom H ) -> H e. Mnd )
54	53	ad2antrr 492	. . . . . . . . 9 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> H e. Mnd )
55	28	adantr 276	. . . . . . . . 9 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( F ` X ) e. ( Base ` H ) )
56	26, 29, 49	mulgnn0p1 13913	. . . . . . . . 9 |- ( ( H e. Mnd /\ m e. NN0 /\ ( F ` X ) e. ( Base ` H ) ) -> ( ( m + 1 ) .X. ( F ` X ) ) = ( ( m .X. ( F ` X ) ) ( +g ` H ) ( F ` X ) ) )
57	54, 36, 55, 56	syl3anc 1278	. . . . . . . 8 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( ( m + 1 ) .X. ( F ` X ) ) = ( ( m .X. ( F ` X ) ) ( +g ` H ) ( F ` X ) ) )
58	52, 57	eqeq12d 2253	. . . . . . 7 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( ( F ` ( ( m + 1 ) .x. X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) <-> ( ( F ` ( m .x. X ) ) ( +g ` H ) ( F ` X ) ) = ( ( m .X. ( F ` X ) ) ( +g ` H ) ( F ` X ) ) ) )
59	33, 58	imbitrrid 156	. . . . . 6 |- ( ( ( F e. ( G MndHom H ) /\ X e. B ) /\ m e. NN0 ) -> ( ( F ` ( m .x. X ) ) = ( m .X. ( F ` X ) ) -> ( F ` ( ( m + 1 ) .x. X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) ) )
60	59	expcom 116	. . . . 5 |- ( m e. NN0 -> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( ( F ` ( m .x. X ) ) = ( m .X. ( F ` X ) ) -> ( F ` ( ( m + 1 ) .x. X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) ) ) )
61	60	a2d 26	. . . 4 |- ( m e. NN0 -> ( ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( m .x. X ) ) = ( m .X. ( F ` X ) ) ) -> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( ( m + 1 ) .x. X ) ) = ( ( m + 1 ) .X. ( F ` X ) ) ) ) )
62	4, 8, 12, 16, 32, 61	nn0ind 9739	. . 3 |- ( N e. NN0 -> ( ( F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) ) )
63	62	3impib 1232	. 2 |- ( ( N e. NN0 /\ F e. ( G MndHom H ) /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )
64	63	3com12 1238	1 |- ( ( F e. ( G MndHom H ) /\ N e. NN0 /\ X e. B ) -> ( F ` ( N .x. X ) ) = ( N .X. ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170   + caddc 8172   NN0cn0 9542   Basecbs 13330   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706   MndHom cmhm 13741  ._gcmg 13899

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-minusg 13786  df-mulg 13900

This theorem is referenced by:  ghmmulg  14036  lgseisenlem4  16106