Theorem mhmima 13265

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 5032	. . 3 |- ( F " X ) C_ ran F
2		eqid 2204	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
3		eqid 2204	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
4	2, 3	mhmf 13239	. . . . 5 |- ( F e. ( M MndHom N ) -> F : ( Base ` M ) --> ( Base ` N ) )
5	4	adantr 276	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> F : ( Base ` M ) --> ( Base ` N ) )
6	5	frnd 5434	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ran F C_ ( Base ` N ) )
7	1, 6	sstrid 3203	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) C_ ( Base ` N ) )
8		eqid 2204	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
9		eqid 2204	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
10	8, 9	mhm0 13242	. . . 4 |- ( F e. ( M MndHom N ) -> ( F ` ( 0g ` M ) ) = ( 0g ` N ) )
11	10	adantr 276	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) = ( 0g ` N ) )
12	5	ffnd 5425	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> F Fn ( Base ` M ) )
13	2	submss 13250	. . . . 5 |- ( X e. (SubMnd ` M ) -> X C_ ( Base ` M ) )
14	13	adantl 277	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> X C_ ( Base ` M ) )
15	8	subm0cl 13252	. . . . 5 |- ( X e. (SubMnd ` M ) -> ( 0g ` M ) e. X )
16	15	adantl 277	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( 0g ` M ) e. X )
17		fnfvima 5818	. . . 4 |- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) /\ ( 0g ` M ) e. X ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) )
18	12, 14, 16, 17	syl3anc 1249	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) )
19	11, 18	eqeltrrd 2282	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( 0g ` N ) e. ( F " X ) )
20		simpll 527	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> F e. ( M MndHom N ) )
21	14	adantr 276	. . . . . . . . . 10 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> X C_ ( Base ` M ) )
22		simprl 529	. . . . . . . . . 10 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> z e. X )
23	21, 22	sseldd 3193	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> z e. ( Base ` M ) )
24		simprr 531	. . . . . . . . . 10 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. X )
25	21, 24	sseldd 3193	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. ( Base ` M ) )
26		eqid 2204	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
27		eqid 2204	. . . . . . . . . 10 |- ( +g ` N ) = ( +g ` N )
28	2, 26, 27	mhmlin 13241	. . . . . . . . 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 ) ) )
29	20, 23, 25, 28	syl3anc 1249	. . . . . . . 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 ) ) )
30	12	adantr 276	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> F Fn ( Base ` M ) )
31	26	submcl 13253	. . . . . . . . . . 11 |- ( ( X e. (SubMnd ` M ) /\ z e. X /\ x e. X ) -> ( z ( +g ` M ) x ) e. X )
32	31	3expb 1206	. . . . . . . . . 10 |- ( ( X e. (SubMnd ` M ) /\ ( z e. X /\ x e. X ) ) -> ( z ( +g ` M ) x ) e. X )
33	32	adantll 476	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( z ( +g ` M ) x ) e. X )
34		fnfvima 5818	. . . . . . . . 9 |- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) /\ ( z ( +g ` M ) x ) e. X ) -> ( F ` ( z ( +g ` M ) x ) ) e. ( F " X ) )
35	30, 21, 33, 34	syl3anc 1249	. . . . . . . 8 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( F ` ( z ( +g ` M ) x ) ) e. ( F " X ) )
36	29, 35	eqeltrrd 2282	. . . . . . 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 ) )
37	36	anassrs 400	. . . . . 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 ) )
38	37	ralrimiva 2578	. . . . 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 ) )
39		oveq2 5951	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( F ` z ) ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) )
40	39	eleq1d 2273	. . . . . . . 8 |- ( y = ( F ` x ) -> ( ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
41	40	ralima 5823	. . . . . . 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 ) ) )
42	12, 14, 41	syl2anc 411	. . . . . 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 ) ) )
43	42	adantr 276	. . . . 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 ) ) )
44	38, 43	mpbird 167	. . . 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 ) )
45	44	ralrimiva 2578	. . 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 ) )
46		oveq1 5950	. . . . . . 7 |- ( x = ( F ` z ) -> ( x ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) y ) )
47	46	eleq1d 2273	. . . . . 6 |- ( x = ( F ` z ) -> ( ( x ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
48	47	ralbidv 2505	. . . . 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 ) ) )
49	48	ralima 5823	. . . 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 ) ) )
50	12, 14, 49	syl2anc 411	. . 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 ) ) )
51	45, 50	mpbird 167	. 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 ) )
52		mhmrcl2 13238	. . . 4 |- ( F e. ( M MndHom N ) -> N e. Mnd )
53	52	adantr 276	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> N e. Mnd )
54	3, 9, 27	issubm 13246	. . 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 ) ) ) )
55	53, 54	syl 14	. 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 ) ) ) )
56	7, 19, 51, 55	mpbir3and 1182	1 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   e. wcel 2175   A.wral 2483   C_ wss 3165   ran crn 4675   "cima 4677   Fn wfn 5265   -->wf 5266   ` cfv 5270  (class class class)co 5943   Basecbs 12774   +g cplusg 12851   0gc0g 13030   Mndcmnd 13190   MndHom cmhm 13231  SubMndcsubmnd 13232

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-map 6736  df-inn 9036  df-ndx 12777  df-slot 12778  df-base 12780  df-mhm 13233  df-submnd 13234

This theorem is referenced by:  rhmima  13955