Theorem mhmima 12875

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 4982	. . 3 |- ( F " X ) C_ ran F
2		eqid 2177	. . . . . 6 |- ( Base ` M ) = ( Base ` M )
3		eqid 2177	. . . . . 6 |- ( Base ` N ) = ( Base ` N )
4	2, 3	mhmf 12856	. . . . 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 5376	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ran F C_ ( Base ` N ) )
7	1, 6	sstrid 3167	. 2 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) C_ ( Base ` N ) )
8		eqid 2177	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
9		eqid 2177	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
10	8, 9	mhm0 12859	. . . 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 5367	. . . 4 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> F Fn ( Base ` M ) )
13	2	submss 12867	. . . . 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 12869	. . . . 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 5752	. . . 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 1238	. . 3 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) )
19	11, 18	eqeltrrd 2255	. 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 3157	. . . . . . . . 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 3157	. . . . . . . . 9 |- ( ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. ( Base ` M ) )
26		eqid 2177	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
27		eqid 2177	. . . . . . . . . 10 |- ( +g ` N ) = ( +g ` N )
28	2, 26, 27	mhmlin 12858	. . . . . . . . 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 1238	. . . . . . . 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 12870	. . . . . . . . . . 11 |- ( ( X e. (SubMnd ` M ) /\ z e. X /\ x e. X ) -> ( z ( +g ` M ) x ) e. X )
32	31	3expb 1204	. . . . . . . . . 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 5752	. . . . . . . . 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 1238	. . . . . . . 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 2255	. . . . . . 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 2550	. . . . 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 5883	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( F ` z ) ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) )
40	39	eleq1d 2246	. . . . . . . 8 |- ( y = ( F ` x ) -> ( ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) )
41	40	ralima 5757	. . . . . . 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 2550	. . 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 5882	. . . . . . 7 |- ( x = ( F ` z ) -> ( x ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) y ) )
47	46	eleq1d 2246	. . . . . 6 |- ( x = ( F ` z ) -> ( ( x ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) )
48	47	ralbidv 2477	. . . . 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 5757	. . . 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 12855	. . . 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 12863	. . 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 1180	1 |- ( ( F e. ( M MndHom N ) /\ X e. (SubMnd ` M ) ) -> ( F " X ) e. (SubMnd ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   C_ wss 3130   ran crn 4628   "cima 4630   Fn wfn 5212   -->wf 5213   ` cfv 5217  (class class class)co 5875   Basecbs 12462   +g cplusg 12536   0gc0g 12705   Mndcmnd 12817   MndHom cmhm 12849  SubMndcsubmnd 12850

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-map 6650  df-inn 8920  df-ndx 12465  df-slot 12466  df-base 12468  df-mhm 12851  df-submnd 12852

This theorem is referenced by: (None)