Theorem resmhm2b 12941

Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypothesis

Ref	Expression
resmhm2.u	|- U = ( T ↾s X )

Assertion

Ref	Expression
resmhm2b	|- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )

Proof of Theorem resmhm2b

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

Step	Hyp	Ref	Expression
1		mhmrcl1 12915	. . . 4 |- ( F e. ( S MndHom T ) -> S e. Mnd )
2	1	adantl 277	. . 3 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> S e. Mnd )
3		resmhm2.u	. . . . 5 |- U = ( T ↾s X )
4	3	submmnd 12932	. . . 4 |- ( X e. (SubMnd ` T ) -> U e. Mnd )
5	4	ad2antrr 488	. . 3 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> U e. Mnd )
6		eqid 2189	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
7		eqid 2189	. . . . . . . . 9 |- ( Base ` T ) = ( Base ` T )
8	6, 7	mhmf 12917	. . . . . . . 8 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
9	8	adantl 277	. . . . . . 7 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
10	9	ffnd 5385	. . . . . 6 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F Fn ( Base ` S ) )
11		simplr 528	. . . . . 6 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ran F C_ X )
12		df-f 5239	. . . . . 6 |- ( F : ( Base ` S ) --> X <-> ( F Fn ( Base ` S ) /\ ran F C_ X ) )
13	10, 11, 12	sylanbrc 417	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F : ( Base ` S ) --> X )
14	3	submbas 12933	. . . . . . 7 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
15	14	ad2antrr 488	. . . . . 6 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> X = ( Base ` U ) )
16	15	feq3d 5373	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F : ( Base ` S ) --> X <-> F : ( Base ` S ) --> ( Base ` U ) ) )
17	13, 16	mpbid 147	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F : ( Base ` S ) --> ( Base ` U ) )
18		eqid 2189	. . . . . . . . 9 |- ( +g ` S ) = ( +g ` S )
19		eqid 2189	. . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
20	6, 18, 19	mhmlin 12919	. . . . . . . 8 |- ( ( F e. ( S MndHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
21	20	3expb 1206	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
22	21	adantll 476	. . . . . 6 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
23	3	a1i 9	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> U = ( T ↾s X ) )
24		eqidd 2190	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` T ) )
25		id 19	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> X e. (SubMnd ` T ) )
26		submrcl 12923	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> T e. Mnd )
27	23, 24, 25, 26	ressplusgd 12640	. . . . . . . 8 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` U ) )
28	27	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( +g ` T ) = ( +g ` U ) )
29	28	oveqd 5913	. . . . . 6 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( ( F ` x ) ( +g ` T ) ( F ` y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
30	22, 29	eqtrd 2222	. . . . 5 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
31	30	ralrimivva 2572	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
32		eqid 2189	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
33		eqid 2189	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
34	32, 33	mhm0 12920	. . . . . 6 |- ( F e. ( S MndHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
35	34	adantl 277	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
36	3, 33	subm0 12934	. . . . . 6 |- ( X e. (SubMnd ` T ) -> ( 0g ` T ) = ( 0g ` U ) )
37	36	ad2antrr 488	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( 0g ` T ) = ( 0g ` U ) )
38	35, 37	eqtrd 2222	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` U ) )
39	17, 31, 38	3jca 1179	. . 3 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F : ( Base ` S ) --> ( Base ` U ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` U ) ) )
40		eqid 2189	. . . 4 |- ( Base ` U ) = ( Base ` U )
41		eqid 2189	. . . 4 |- ( +g ` U ) = ( +g ` U )
42		eqid 2189	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
43	6, 40, 18, 41, 32, 42	ismhm 12913	. . 3 |- ( F e. ( S MndHom U ) <-> ( ( S e. Mnd /\ U e. Mnd ) /\ ( F : ( Base ` S ) --> ( Base ` U ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` U ) ) ) )
44	2, 5, 39, 43	syl21anbrc 1184	. 2 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F e. ( S MndHom U ) )
45	3	resmhm2 12940	. . . 4 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )
46	45	ancoms 268	. . 3 |- ( ( X e. (SubMnd ` T ) /\ F e. ( S MndHom U ) ) -> F e. ( S MndHom T ) )
47	46	adantlr 477	. 2 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom U ) ) -> F e. ( S MndHom T ) )
48	44, 47	impbida 596	1 |- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   C_ wss 3144   ran crn 4645   Fn wfn 5230   -->wf 5231   ` cfv 5235  (class class class)co 5896   Basecbs 12512   ↾_s cress 12513   +g cplusg 12589   0gc0g 12761   Mndcmnd 12877   MndHom cmhm 12909  SubMndcsubmnd 12910

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-1re 7935  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-i2m1 7946  ax-0lt1 7947  ax-0id 7949  ax-rnegex 7950  ax-pre-ltirr 7953  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-1st 6165  df-2nd 6166  df-map 6676  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-inn 8950  df-2 9008  df-ndx 12515  df-slot 12516  df-base 12518  df-sets 12519  df-iress 12520  df-plusg 12602  df-0g 12763  df-mgm 12832  df-sgrp 12865  df-mnd 12878  df-mhm 12911  df-submnd 12912

This theorem is referenced by:  resghm2b  13201