Theorem resmhm2b 13064

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 13038	. . . 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 13055	. . . 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 2193	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
7		eqid 2193	. . . . . . . . 9 |- ( Base ` T ) = ( Base ` T )
8	6, 7	mhmf 13040	. . . . . . . 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 5405	. . . . . 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 5259	. . . . . 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 13056	. . . . . . 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 5393	. . . . 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 2193	. . . . . . . . 9 |- ( +g ` S ) = ( +g ` S )
19		eqid 2193	. . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
20	6, 18, 19	mhmlin 13042	. . . . . . . 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 2194	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` T ) )
25		id 19	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> X e. (SubMnd ` T ) )
26		submrcl 13046	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> T e. Mnd )
27	23, 24, 25, 26	ressplusgd 12749	. . . . . . . 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 5936	. . . . . 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 2226	. . . . 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 2576	. . . 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 2193	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
33		eqid 2193	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
34	32, 33	mhm0 13043	. . . . . 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 13057	. . . . . 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 2226	. . . 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 2193	. . . 4 |- ( Base ` U ) = ( Base ` U )
41		eqid 2193	. . . 4 |- ( +g ` U ) = ( +g ` U )
42		eqid 2193	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
43	6, 40, 18, 41, 32, 42	ismhm 13036	. . 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 13063	. . . 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 2164   A.wral 2472   C_ wss 3154   ran crn 4661   Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5919   Basecbs 12621   ↾_s cress 12622   +g cplusg 12698   0gc0g 12870   Mndcmnd 13000   MndHom cmhm 13032  SubMndcsubmnd 13033

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-mhm 13034  df-submnd 13035

This theorem is referenced by:  resghm2b  13335  resrhm2b  13748  lgseisenlem4  15230