Theorem resmhm2 12963

Description: One direction of resmhm2b 12964. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypothesis

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

Assertion

Ref	Expression
resmhm2	|- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )

Proof of Theorem resmhm2

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

Step	Hyp	Ref	Expression
1		mhmrcl1 12938	. . 3 |- ( F e. ( S MndHom U ) -> S e. Mnd )
2		submrcl 12946	. . 3 |- ( X e. (SubMnd ` T ) -> T e. Mnd )
3	1, 2	anim12i 338	. 2 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( S e. Mnd /\ T e. Mnd ) )
4		eqid 2189	. . . . 5 |- ( Base ` S ) = ( Base ` S )
5		eqid 2189	. . . . 5 |- ( Base ` U ) = ( Base ` U )
6	4, 5	mhmf 12940	. . . 4 |- ( F e. ( S MndHom U ) -> F : ( Base ` S ) --> ( Base ` U ) )
7		resmhm2.u	. . . . . 6 |- U = ( T ↾s X )
8	7	submbas 12956	. . . . 5 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
9		eqid 2189	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
10	9	submss 12951	. . . . 5 |- ( X e. (SubMnd ` T ) -> X C_ ( Base ` T ) )
11	8, 10	eqsstrrd 3207	. . . 4 |- ( X e. (SubMnd ` T ) -> ( Base ` U ) C_ ( Base ` T ) )
12		fss 5399	. . . 4 |- ( ( F : ( Base ` S ) --> ( Base ` U ) /\ ( Base ` U ) C_ ( Base ` T ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
13	6, 11, 12	syl2an 289	. . 3 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
14		eqid 2189	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
15		eqid 2189	. . . . . . . 8 |- ( +g ` U ) = ( +g ` U )
16	4, 14, 15	mhmlin 12942	. . . . . . 7 |- ( ( F e. ( S MndHom U ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
17	16	3expb 1206	. . . . . 6 |- ( ( F e. ( S MndHom U ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
18	17	adantlr 477	. . . . 5 |- ( ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
19	7	a1i 9	. . . . . . . 8 |- ( X e. (SubMnd ` T ) -> U = ( T ↾s X ) )
20		eqidd 2190	. . . . . . . 8 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` T ) )
21		id 19	. . . . . . . 8 |- ( X e. (SubMnd ` T ) -> X e. (SubMnd ` T ) )
22	19, 20, 21, 2	ressplusgd 12651	. . . . . . 7 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` U ) )
23	22	ad2antlr 489	. . . . . 6 |- ( ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( +g ` T ) = ( +g ` U ) )
24	23	oveqd 5917	. . . . 5 |- ( ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( ( F ` x ) ( +g ` T ) ( F ` y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
25	18, 24	eqtr4d 2225	. . . 4 |- ( ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
26	25	ralrimivva 2572	. . 3 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
27		eqid 2189	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
28		eqid 2189	. . . . . 6 |- ( 0g ` U ) = ( 0g ` U )
29	27, 28	mhm0 12943	. . . . 5 |- ( F e. ( S MndHom U ) -> ( F ` ( 0g ` S ) ) = ( 0g ` U ) )
30	29	adantr 276	. . . 4 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` U ) )
31		eqid 2189	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
32	7, 31	subm0 12957	. . . . 5 |- ( X e. (SubMnd ` T ) -> ( 0g ` T ) = ( 0g ` U ) )
33	32	adantl 277	. . . 4 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( 0g ` T ) = ( 0g ` U ) )
34	30, 33	eqtr4d 2225	. . 3 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
35	13, 26, 34	3jca 1179	. 2 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) )
36		eqid 2189	. . 3 |- ( +g ` T ) = ( +g ` T )
37	4, 9, 14, 36, 27, 31	ismhm 12936	. 2 |- ( F e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( F : ( Base ` S ) --> ( Base ` T ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` T ) ) ) )
38	3, 35, 37	sylanbrc 417	1 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2160   A.wral 2468   C_ wss 3144   -->wf 5234   ` cfv 5238  (class class class)co 5900   Basecbs 12523   ↾_s cress 12524   +g cplusg 12600   0gc0g 12772   Mndcmnd 12900   MndHom cmhm 12932  SubMndcsubmnd 12933

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 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-ltadd 7962

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 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-map 6680  df-pnf 8029  df-mnf 8030  df-ltxr 8032  df-inn 8955  df-2 9013  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-0g 12774  df-mgm 12843  df-sgrp 12888  df-mnd 12901  df-mhm 12934  df-submnd 12935

This theorem is referenced by:  resmhm2b  12964  resghm2  13225