Theorem resmhm2b 13191

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 13165	. . . 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 13182	. . . 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 2196	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
7		eqid 2196	. . . . . . . . 9 |- ( Base ` T ) = ( Base ` T )
8	6, 7	mhmf 13167	. . . . . . . 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 5411	. . . . . 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 5263	. . . . . 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 13183	. . . . . . 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 5399	. . . . 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 2196	. . . . . . . . 9 |- ( +g ` S ) = ( +g ` S )
19		eqid 2196	. . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
20	6, 18, 19	mhmlin 13169	. . . . . . . 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 2197	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` T ) )
25		id 19	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> X e. (SubMnd ` T ) )
26		submrcl 13173	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> T e. Mnd )
27	23, 24, 25, 26	ressplusgd 12831	. . . . . . . 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 5942	. . . . . 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 2229	. . . . 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 2579	. . . 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 2196	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
33		eqid 2196	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
34	32, 33	mhm0 13170	. . . . . 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 13184	. . . . . 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 2229	. . . 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 2196	. . . 4 |- ( Base ` U ) = ( Base ` U )
41		eqid 2196	. . . 4 |- ( +g ` U ) = ( +g ` U )
42		eqid 2196	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
43	6, 40, 18, 41, 32, 42	ismhm 13163	. . 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 13190	. . . 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 2167   A.wral 2475   C_ wss 3157   ran crn 4665   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5925   Basecbs 12703   ↾_s cress 12704   +g cplusg 12780   0gc0g 12958   Mndcmnd 13118   MndHom cmhm 13159  SubMndcsubmnd 13160

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-pre-ltirr 8008  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-map 6718  df-pnf 8080  df-mnf 8081  df-ltxr 8083  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-mhm 13161  df-submnd 13162

This theorem is referenced by:  resghm2b  13468  resrhm2b  13881  lgseisenlem4  15398