Theorem resmhm2 13772

Description: One direction of resmhm2b 13773. (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 13747	. . 3 |- ( F e. ( S MndHom U ) -> S e. Mnd )
2		submrcl 13755	. . 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 2238	. . . . 5 |- ( Base ` S ) = ( Base ` S )
5		eqid 2238	. . . . 5 |- ( Base ` U ) = ( Base ` U )
6	4, 5	mhmf 13749	. . . 4 |- ( F e. ( S MndHom U ) -> F : ( Base ` S ) --> ( Base ` U ) )
7		resmhm2.u	. . . . . 6 |- U = ( T ↾s X )
8	7	submbas 13765	. . . . 5 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
9		eqid 2238	. . . . . 6 |- ( Base ` T ) = ( Base ` T )
10	9	submss 13760	. . . . 5 |- ( X e. (SubMnd ` T ) -> X C_ ( Base ` T ) )
11	8, 10	eqsstrrd 3285	. . . 4 |- ( X e. (SubMnd ` T ) -> ( Base ` U ) C_ ( Base ` T ) )
12		fss 5541	. . . 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 2238	. . . . . . . 8 |- ( +g ` S ) = ( +g ` S )
15		eqid 2238	. . . . . . . 8 |- ( +g ` U ) = ( +g ` U )
16	4, 14, 15	mhmlin 13751	. . . . . . 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 1235	. . . . . 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 481	. . . . 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 2239	. . . . . . . 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 13460	. . . . . . 7 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` U ) )
23	22	ad2antlr 493	. . . . . 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 6092	. . . . 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 2274	. . . 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 2632	. . 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 2238	. . . . . 6 |- ( 0g ` S ) = ( 0g ` S )
28		eqid 2238	. . . . . 6 |- ( 0g ` U ) = ( 0g ` U )
29	27, 28	mhm0 13752	. . . . 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 2238	. . . . . 6 |- ( 0g ` T ) = ( 0g ` T )
32	7, 31	subm0 13766	. . . . 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 2274	. . 3 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
35	13, 26, 34	3jca 1208	. 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 2238	. . 3 |- ( +g ` T ) = ( +g ` T )
37	4, 9, 14, 36, 27, 31	ismhm 13745	. 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 421	1 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   -->wf 5368   ` cfv 5372  (class class class)co 6075   Basecbs 13330   ↾_s cress 13331   +g cplusg 13408   0gc0g 13587   Mndcmnd 13706   MndHom cmhm 13741  SubMndcsubmnd 13742

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-ltxr 8355  df-inn 9284  df-2 9342  df-ndx 13333  df-slot 13334  df-base 13336  df-sets 13337  df-iress 13338  df-plusg 13421  df-0g 13589  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743  df-submnd 13744

This theorem is referenced by:  resmhm2b  13773  resghm2  14041  lgseisenlem4  16106