Theorem mhmima 17363

Description: The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)

Assertion

Ref	Expression
mhmima	|- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )

Proof of Theorem mhmima

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 5477	. . 3 |- ( F " X ) C_ ran F
2		eqid 2622	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
3		eqid 2622	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
4	2, 3	mhmf 17340	. . . . 5 |- ( F e. ( M MndHom N ) -> F : ( Base ` M ) --> ( Base ` N ) )
5	4	adantr 481	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> F : ( Base ` M ) --> ( Base ` N ) )
6		frn 6053	. . . 4 |- ( F : ( Base ` M ) --> ( Base ` N ) -> ran F C_ ( Base ` N ) )
7	5, 6	syl 17	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ran F C_ ( Base ` N ) )
8	1, 7	syl5ss 3614	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) C_ ( Base ` N ) )
9		eqid 2622	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
10		eqid 2622	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
11	9, 10	mhm0 17343	. . . 4 |- ( F e. ( M MndHom N ) -> ( F ` ( 0g ` M ) ) = ( 0g ` N ) )
12	11	adantr 481	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) = ( 0g ` N ) )
13		ffn 6045	. . . . 5 |- ( F : ( Base ` M ) --> ( Base ` N ) -> F Fn ( Base ` M ) )
14	5, 13	syl 17	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> F Fn ( Base ` M ) )
15	2	submss 17350	. . . . 5 |- ( X e. (SubMnd ` M ) -> X C_ ( Base ` M ) )
16	15	adantl 482	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> X C_ ( Base ` M ) )
17	9	subm0cl 17352	. . . . 5 |- ( X e. (SubMnd ` M ) -> ( 0g ` M ) e. X )
18	17	adantl 482	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( 0g ` M ) e. X )
19		fnfvima 6496	. . . 4 |- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) /\ ( 0g ` M ) e. X ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) )
20	14, 16, 18, 19	syl3anc 1326	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) )
21	12, 20	eqeltrrd 2702	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( 0g ` N ) e. ( F " X ) )
22		simpll 790	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> F e. ( M MndHom N ) )
23	16	adantr 481	. . . . . . . . . 10 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> X C_ ( Base ` M ) )
24		simprl 794	. . . . . . . . . 10 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> z e. X )
25	23, 24	sseldd 3604	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> z e. ( Base ` M ) )
26		simprr 796	. . . . . . . . . 10 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. X )
27	23, 26	sseldd 3604	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. ( Base ` M ) )
28		eqid 2622	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
29		eqid 2622	. . . . . . . . . 10 |- ( +g ` N ) = ( +g ` N )
30	2, 28, 29	mhmlin 17342	. . . . . . . . 9 |- ( ( F e. ( M MndHom N ) /\ z e. ( Base ` M ) /\ x e. ( Base ` M ) ) -> ( F ` ( z ( +g ` M ) x ) ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) )
31	22, 25, 27, 30	syl3anc 1326	. . . . . . . 8 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( F ` ( z ( +g ` M ) x ) ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) )
32	14	adantr 481	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> F Fn ( Base ` M ) )
33	28	submcl 17353	. . . . . . . . . . 11 |- ( ( X e. (SubMnd ` M ) /\ z e. X /\ x e. X ) -> ( z ( +g ` M ) x ) e. X )
34	33	3expb 1266	. . . . . . . . . 10 |- ( ( X e. (SubMnd ` M ) /\ ( z e. X /\ x e. X ) ) -> ( z ( +g ` M ) x ) e. X )
35	34	adantll 750	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( z ( +g ` M ) x ) e. X )
36		fnfvima 6496	. . . . . . . . 9 |- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) /\ ( z ( +g ` M ) x ) e. X ) -> ( F ` ( z ( +g ` M ) x ) ) e. ( F " X ) )
37	32, 23, 35, 36	syl3anc 1326	. . . . . . . 8 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( F ` ( z ( +g ` M ) x ) ) e. ( F " X ) )
38	31, 37	eqeltrrd 2702	. . . . . . 7 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) )
39	38	anassrs 680	. . . . . 6 |- ( ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ z e. X ) /\ x e. X ) -> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) )
40	39	ralrimiva 2966	. . . . 5 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ z e. X ) -> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) )
41		oveq2 6658	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( F ` z ) ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) )
42	41	eleq1d 2686	. . . . . . . 8 |- ( y = ( F ` x ) -> ( ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
43	42	ralima 6498	. . . . . . 7 |- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) ) -> ( A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
44	14, 16, 43	syl2anc 693	. . . . . 6 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
45	44	adantr 481	. . . . 5 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ z e. X ) -> ( A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
46	40, 45	mpbird 247	. . . 4 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ z e. X ) -> A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) )
47	46	ralrimiva 2966	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> A. z e. X A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) )
48		oveq1 6657	. . . . . . 7 |- ( x = ( F ` z ) -> ( x ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) y ) )
49	48	eleq1d 2686	. . . . . 6 |- ( x = ( F ` z ) -> ( ( x ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
50	49	ralbidv 2986	. . . . 5 |- ( x = ( F ` z ) -> ( A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) <-> A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
51	50	ralima 6498	. . . 4 |- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) ) -> ( A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) <-> A. z e. X A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
52	14, 16, 51	syl2anc 693	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) <-> A. z e. X A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
53	47, 52	mpbird 247	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) )
54		mhmrcl2 17339	. . . 4 |- ( F e. ( M MndHom N ) -> N e. Mnd )
55	54	adantr 481	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> N e. Mnd )
56	3, 10, 29	issubm 17347	. . 3 |- ( N e. Mnd -> ( ( F " X ) e. (SubMnd ` N ) <-> ( ( F " X ) C_ ( Base ` N ) /\ ( 0g ` N ) e. ( F " X ) /\ A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) ) ) )
57	55, 56	syl 17	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( ( F " X ) e. (SubMnd ` N ) <-> ( ( F " X ) C_ ( Base ` N ) /\ ( 0g ` N ) e. ( F " X ) /\ A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) ) ) )
58	8, 21, 53, 57	mpbir3and 1245	1 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294   MndHom cmhm 17333  SubMndcsubmnd 17334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336

This theorem is referenced by:  rhmima  18811