Theorem resmhm2 13435

Description: One direction of resmhm2b 13436. (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 13410	. . 3 |- ( F e. ( S MndHom U ) -> S e. Mnd )
2		submrcl 13418	. . 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 2207	. . . . 5 |- ( Base ` S ) = ( Base ` S )
5		eqid 2207	. . . . 5 |- ( Base ` U ) = ( Base ` U )
6	4, 5	mhmf 13412	. . . 4 |- ( F e. ( S MndHom U ) -> F : ( Base ` S ) --> ( Base ` U ) )
7		resmhm2.u	. . . . . 6 |- U = ( T ↾s X )
8	7	submbas 13428	. . . . 5 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
9		eqid 2207	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
10	9	submss 13423	. . . . 5 |- ( X e. (SubMnd ` T ) -> X C_ ( Base ` T ) )
11	8, 10	eqsstrrd 3238	. . . 4 |- ( X e. (SubMnd ` T ) -> ( Base ` U ) C_ ( Base ` T ) )
12		fss 5457	. . . 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 2207	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
15		eqid 2207	. . . . . . . 8 |- ( +g ` U ) = ( +g ` U )
16	4, 14, 15	mhmlin 13414	. . . . . . 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 1207	. . . . . 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 2208	. . . . . . . 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 13076	. . . . . . 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 5984	. . . . 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 2243	. . . 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 2590	. . 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 2207	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
28		eqid 2207	. . . . . 6 |- ( 0g ` U ) = ( 0g ` U )
29	27, 28	mhm0 13415	. . . . 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 2207	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
32	7, 31	subm0 13429	. . . . 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 2243	. . 3 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
35	13, 26, 34	3jca 1180	. 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 2207	. . 3 |- ( +g ` T ) = ( +g ` T )
37	4, 9, 14, 36, 27, 31	ismhm 13408	. 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 981   = wceq 1373   e. wcel 2178   A.wral 2486   C_ wss 3174   -->wf 5286   ` cfv 5290  (class class class)co 5967   Basecbs 12947   ↾_s cress 12948   +g cplusg 13024   0gc0g 13203   Mndcmnd 13363   MndHom cmhm 13404  SubMndcsubmnd 13405

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-mhm 13406  df-submnd 13407

This theorem is referenced by:  resmhm2b  13436  resghm2  13712  lgseisenlem4  15665