Theorem resmhm2 13634

Description: One direction of resmhm2b 13635. (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 13609	. . 3 |- ( F e. ( S MndHom U ) -> S e. Mnd )
2		submrcl 13617	. . 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 2231	. . . . 5 |- ( Base ` S ) = ( Base ` S )
5		eqid 2231	. . . . 5 |- ( Base ` U ) = ( Base ` U )
6	4, 5	mhmf 13611	. . . 4 |- ( F e. ( S MndHom U ) -> F : ( Base ` S ) --> ( Base ` U ) )
7		resmhm2.u	. . . . . 6 |- U = ( T ↾s X )
8	7	submbas 13627	. . . . 5 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
9		eqid 2231	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
10	9	submss 13622	. . . . 5 |- ( X e. (SubMnd ` T ) -> X C_ ( Base ` T ) )
11	8, 10	eqsstrrd 3265	. . . 4 |- ( X e. (SubMnd ` T ) -> ( Base ` U ) C_ ( Base ` T ) )
12		fss 5501	. . . 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 2231	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
15		eqid 2231	. . . . . . . 8 |- ( +g ` U ) = ( +g ` U )
16	4, 14, 15	mhmlin 13613	. . . . . . 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 1231	. . . . . 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 2232	. . . . . . . 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 13275	. . . . . . 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 6045	. . . . 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 2267	. . . 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 2615	. . 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 2231	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
28		eqid 2231	. . . . . 6 |- ( 0g ` U ) = ( 0g ` U )
29	27, 28	mhm0 13614	. . . . 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 2231	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
32	7, 31	subm0 13628	. . . . 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 2267	. . 3 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
35	13, 26, 34	3jca 1204	. 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 2231	. . 3 |- ( +g ` T ) = ( +g ` T )
37	4, 9, 14, 36, 27, 31	ismhm 13607	. 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 1005   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   -->wf 5329   ` cfv 5333  (class class class)co 6028   Basecbs 13145   ↾_s cress 13146   +g cplusg 13223   0gc0g 13402   Mndcmnd 13562   MndHom cmhm 13603  SubMndcsubmnd 13604

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-ltadd 8191

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-mhm 13605  df-submnd 13606

This theorem is referenced by:  resmhm2b  13635  resghm2  13911  lgseisenlem4  15875