Theorem resmhm2b 13562

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 13536	. . . 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 13553	. . . 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 2229	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
7		eqid 2229	. . . . . . . . 9 |- ( Base ` T ) = ( Base ` T )
8	6, 7	mhmf 13538	. . . . . . . 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 5480	. . . . . 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 5328	. . . . . 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 13554	. . . . . . 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 5468	. . . . 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 2229	. . . . . . . . 9 |- ( +g ` S ) = ( +g ` S )
19		eqid 2229	. . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
20	6, 18, 19	mhmlin 13540	. . . . . . . 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 1228	. . . . . . 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 2230	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` T ) )
25		id 19	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> X e. (SubMnd ` T ) )
26		submrcl 13544	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> T e. Mnd )
27	23, 24, 25, 26	ressplusgd 13202	. . . . . . . 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 6030	. . . . . 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 2262	. . . . 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 2612	. . . 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 2229	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
33		eqid 2229	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
34	32, 33	mhm0 13541	. . . . . 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 13555	. . . . . 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 2262	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` U ) )
39	17, 31, 38	3jca 1201	. . 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 2229	. . . 4 |- ( Base ` U ) = ( Base ` U )
41		eqid 2229	. . . 4 |- ( +g ` U ) = ( +g ` U )
42		eqid 2229	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
43	6, 40, 18, 41, 32, 42	ismhm 13534	. . 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 1206	. 2 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F e. ( S MndHom U ) )
45	3	resmhm2 13561	. . . 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 598	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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   C_ wss 3198   ran crn 4724   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   Basecbs 13072   ↾_s cress 13073   +g cplusg 13150   0gc0g 13329   Mndcmnd 13489   MndHom cmhm 13530  SubMndcsubmnd 13531

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-map 6814  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mhm 13532  df-submnd 13533

This theorem is referenced by:  resghm2b  13839  resrhm2b  14253  lgseisenlem4  15792