Theorem mhmima 13438

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 5052	. . 3 |- ( F " X ) C_ ran F
2		eqid 2207	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
3		eqid 2207	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
4	2, 3	mhmf 13412	. . . . 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 5455	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ran F C_ ( Base ` N ) )
7	1, 6	sstrid 3212	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) C_ ( Base ` N ) )
8		eqid 2207	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
9		eqid 2207	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
10	8, 9	mhm0 13415	. . . 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 5446	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> F Fn ( Base ` M ) )
13	2	submss 13423	. . . . 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 13425	. . . . 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 5842	. . . 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 1250	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) )
19	11, 18	eqeltrrd 2285	. 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 3202	. . . . . . . . 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 3202	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. ( Base ` M ) )
26		eqid 2207	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
27		eqid 2207	. . . . . . . . . 10 |- ( +g ` N ) = ( +g ` N )
28	2, 26, 27	mhmlin 13414	. . . . . . . . 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 1250	. . . . . . . 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 13426	. . . . . . . . . . 11 |- ( ( X e. (SubMnd ` M ) /\ z e. X /\ x e. X ) -> ( z ( +g ` M ) x ) e. X )
32	31	3expb 1207	. . . . . . . . . 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 5842	. . . . . . . . 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 1250	. . . . . . . 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 2285	. . . . . . 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 2581	. . . . 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 5975	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( F ` z ) ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) )
40	39	eleq1d 2276	. . . . . . . 8 |- ( y = ( F ` x ) -> ( ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
41	40	ralima 5847	. . . . . . 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 2581	. . 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 5974	. . . . . . 7 |- ( x = ( F ` z ) -> ( x ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) y ) )
47	46	eleq1d 2276	. . . . . 6 |- ( x = ( F ` z ) -> ( ( x ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
48	47	ralbidv 2508	. . . . 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 5847	. . . 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 13411	. . . 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 13419	. . 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 1183	1 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   ran crn 4694   "cima 4696   Fn wfn 5285   -->wf 5286   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   0gc0g 13203   Mndcmnd 13363   MndHom cmhm 13404  SubMndcsubmnd 13405

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-inn 9072  df-ndx 12950  df-slot 12951  df-base 12953  df-mhm 13406  df-submnd 13407

This theorem is referenced by:  rhmima  14128