Theorem resmhm2b 13773

Description: Restriction of the codomain of a homomorphism. (Contributed by Mario Carneiro, 18-Jun-2015.)

Hypothesis

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

Assertion

Ref	Expression
resmhm2b	|- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )

Proof of Theorem resmhm2b

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

Step	Hyp	Ref	Expression
1		mhmrcl1 13747	. . . 4 |- ( F e. ( S MndHom T ) -> S e. Mnd )
2	1	adantl 277	. . 3 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> S e. Mnd )
3		resmhm2.u	. . . . 5 |- U = ( T ↾s X )
4	3	submmnd 13764	. . . 4 |- ( X e. (SubMnd ` T ) -> U e. Mnd )
5	4	ad2antrr 492	. . 3 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> U e. Mnd )
6		eqid 2238	. . . . . . . . 9 |- ( Base ` S ) = ( Base ` S )
7		eqid 2238	. . . . . . . . 9 |- ( Base ` T ) = ( Base ` T )
8	6, 7	mhmf 13749	. . . . . . . 8 |- ( F e. ( S MndHom T ) -> F : ( Base ` S ) --> ( Base ` T ) )
9	8	adantl 277	. . . . . . 7 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F : ( Base ` S ) --> ( Base ` T ) )
10	9	ffnd 5529	. . . . . 6 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F Fn ( Base ` S ) )
11		simplr 533	. . . . . 6 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ran F C_ X )
12		df-f 5376	. . . . . 6 |- ( F : ( Base ` S ) --> X <-> ( F Fn ( Base ` S ) /\ ran F C_ X ) )
13	10, 11, 12	sylanbrc 421	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F : ( Base ` S ) --> X )
14	3	submbas 13765	. . . . . . 7 |- ( X e. (SubMnd ` T ) -> X = ( Base ` U ) )
15	14	ad2antrr 492	. . . . . 6 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> X = ( Base ` U ) )
16	15	feq3d 5517	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F : ( Base ` S ) --> X <-> F : ( Base ` S ) --> ( Base ` U ) ) )
17	13, 16	mpbid 147	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F : ( Base ` S ) --> ( Base ` U ) )
18		eqid 2238	. . . . . . . . 9 |- ( +g ` S ) = ( +g ` S )
19		eqid 2238	. . . . . . . . 9 |- ( +g ` T ) = ( +g ` T )
20	6, 18, 19	mhmlin 13751	. . . . . . . 8 |- ( ( F e. ( S MndHom T ) /\ x e. ( Base ` S ) /\ y e. ( Base ` S ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
21	20	3expb 1235	. . . . . . 7 |- ( ( F e. ( S MndHom T ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
22	21	adantll 480	. . . . . 6 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` T ) ( F ` y ) ) )
23	3	a1i 9	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> U = ( T ↾s X ) )
24		eqidd 2239	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` T ) )
25		id 19	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> X e. (SubMnd ` T ) )
26		submrcl 13755	. . . . . . . . 9 |- ( X e. (SubMnd ` T ) -> T e. Mnd )
27	23, 24, 25, 26	ressplusgd 13460	. . . . . . . 8 |- ( X e. (SubMnd ` T ) -> ( +g ` T ) = ( +g ` U ) )
28	27	ad3antrrr 496	. . . . . . 7 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( +g ` T ) = ( +g ` U ) )
29	28	oveqd 6092	. . . . . 6 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( ( F ` x ) ( +g ` T ) ( F ` y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
30	22, 29	eqtrd 2271	. . . . 5 |- ( ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) /\ ( x e. ( Base ` S ) /\ y e. ( Base ` S ) ) ) -> ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
31	30	ralrimivva 2632	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) )
32		eqid 2238	. . . . . . 7 |- ( 0g ` S ) = ( 0g ` S )
33		eqid 2238	. . . . . . 7 |- ( 0g ` T ) = ( 0g ` T )
34	32, 33	mhm0 13752	. . . . . 6 |- ( F e. ( S MndHom T ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
35	34	adantl 277	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` T ) )
36	3, 33	subm0 13766	. . . . . 6 |- ( X e. (SubMnd ` T ) -> ( 0g ` T ) = ( 0g ` U ) )
37	36	ad2antrr 492	. . . . 5 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( 0g ` T ) = ( 0g ` U ) )
38	35, 37	eqtrd 2271	. . . 4 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F ` ( 0g ` S ) ) = ( 0g ` U ) )
39	17, 31, 38	3jca 1208	. . 3 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> ( F : ( Base ` S ) --> ( Base ` U ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` U ) ) )
40		eqid 2238	. . . 4 |- ( Base ` U ) = ( Base ` U )
41		eqid 2238	. . . 4 |- ( +g ` U ) = ( +g ` U )
42		eqid 2238	. . . 4 |- ( 0g ` U ) = ( 0g ` U )
43	6, 40, 18, 41, 32, 42	ismhm 13745	. . 3 |- ( F e. ( S MndHom U ) <-> ( ( S e. Mnd /\ U e. Mnd ) /\ ( F : ( Base ` S ) --> ( Base ` U ) /\ A. x e. ( Base ` S ) A. y e. ( Base ` S ) ( F ` ( x ( +g ` S ) y ) ) = ( ( F ` x ) ( +g ` U ) ( F ` y ) ) /\ ( F ` ( 0g ` S ) ) = ( 0g ` U ) ) ) )
44	2, 5, 39, 43	syl21anbrc 1213	. 2 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom T ) ) -> F e. ( S MndHom U ) )
45	3	resmhm2 13772	. . . 4 |- ( ( F e. ( S MndHom U ) /\ X e. (SubMnd ` T ) ) -> F e. ( S MndHom T ) )
46	45	ancoms 268	. . 3 |- ( ( X e. (SubMnd ` T ) /\ F e. ( S MndHom U ) ) -> F e. ( S MndHom T ) )
47	46	adantlr 481	. 2 |- ( ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) /\ F e. ( S MndHom U ) ) -> F e. ( S MndHom T ) )
48	44, 47	impbida 604	1 |- ( ( X e. (SubMnd ` T ) /\ ran F C_ X ) -> ( F e. ( S MndHom T ) <-> F e. ( S MndHom U ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   C_ wss 3220   ran crn 4770   Fn wfn 5367   -->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:  resghm2b  14042  resrhm2b  14530  lgseisenlem4  16106