Theorem resmhm2 13570

Description: One direction of resmhm2b 13571. (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 13545	. . 3 |- ( F e. ( S MndHom U ) -> S e. Mnd )
2		submrcl 13553	. . 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 13547	. . . 4 |- ( F e. ( S MndHom U ) -> F : ( Base ` S ) --> ( Base ` U ) )
7		resmhm2.u	. . . . . 6 |- U = ( T ↾s X )
8	7	submbas 13563	. . . . 5 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
9		eqid 2231	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
10	9	submss 13558	. . . . 5 |- ( X e. (SubMnd ` T ) -> X C_ ( Base ` T ) )
11	8, 10	eqsstrrd 3264	. . . 4 |- ( X e. (SubMnd ` T ) -> ( Base ` U ) C_ ( Base ` T ) )
12		fss 5494	. . . 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 13549	. . . . . . 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 1230	. . . . . 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 13211	. . . . . . 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 6034	. . . . 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 2614	. . 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 13550	. . . . 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 13564	. . . . 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 1203	. 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 13543	. 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 1004   = wceq 1397   e. wcel 2202   A.wral 2510   C_ wss 3200   -->wf 5322   ` cfv 5326  (class class class)co 6017   Basecbs 13081   ↾_s cress 13082   +g cplusg 13159   0gc0g 13338   Mndcmnd 13498   MndHom cmhm 13539  SubMndcsubmnd 13540

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mhm 13541  df-submnd 13542

This theorem is referenced by:  resmhm2b  13571  resghm2  13847  lgseisenlem4  15801