Theorem pwssplit3 19061

Description: Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Hypotheses

Ref	Expression
pwssplit1.y	|- Y = ( W ^s U )
pwssplit1.z	|- Z = ( W ^s V )
pwssplit1.b	|- B = ( Base ` Y )
pwssplit1.c	|- C = ( Base ` Z )
pwssplit1.f	|- F = ( x e. B |-> ( x |` V ) )

Assertion

Ref	Expression
pwssplit3	|- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> F e. ( Y LMHom Z ) )

Distinct variable groups:   x, Y   x, W   x, U   x, Z   x, V   x, B   x, C   x, X

Allowed substitution hint:   F( x)

Proof of Theorem pwssplit3

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pwssplit1.b	. 2 |- B = ( Base ` Y )
2		eqid 2622	. 2 |- ( .s ` Y ) = ( .s ` Y )
3		eqid 2622	. 2 |- ( .s ` Z ) = ( .s ` Z )
4		eqid 2622	. 2 |- (Scalar ` Y ) = (Scalar ` Y )
5		eqid 2622	. 2 |- (Scalar ` Z ) = (Scalar ` Z )
6		eqid 2622	. 2 |- ( Base ` (Scalar ` Y ) ) = ( Base ` (Scalar ` Y ) )
7		simp1 1061	. . 3 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> W e. LMod )
8		simp2 1062	. . 3 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> U e. X )
9		pwssplit1.y	. . . 4 |- Y = ( W ^s U )
10	9	pwslmod 18970	. . 3 |- ( ( W e. LMod /\ U e. X ) -> Y e. LMod )
11	7, 8, 10	syl2anc 693	. 2 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> Y e. LMod )
12		simp3 1063	. . . 4 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> V C_ U )
13	8, 12	ssexd 4805	. . 3 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> V e. _V )
14		pwssplit1.z	. . . 4 |- Z = ( W ^s V )
15	14	pwslmod 18970	. . 3 |- ( ( W e. LMod /\ V e. _V ) -> Z e. LMod )
16	7, 13, 15	syl2anc 693	. 2 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> Z e. LMod )
17		eqid 2622	. . . . 5 |- (Scalar ` W ) = (Scalar ` W )
18	14, 17	pwssca 16156	. . . 4 |- ( ( W e. LMod /\ V e. _V ) -> (Scalar ` W ) = (Scalar ` Z ) )
19	7, 13, 18	syl2anc 693	. . 3 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> (Scalar ` W ) = (Scalar ` Z ) )
20	9, 17	pwssca 16156	. . . 4 |- ( ( W e. LMod /\ U e. X ) -> (Scalar ` W ) = (Scalar ` Y ) )
21	7, 8, 20	syl2anc 693	. . 3 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> (Scalar ` W ) = (Scalar ` Y ) )
22	19, 21	eqtr3d 2658	. 2 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> (Scalar ` Z ) = (Scalar ` Y ) )
23		lmodgrp 18870	. . 3 |- ( W e. LMod -> W e. Grp )
24		pwssplit1.c	. . . 4 |- C = ( Base ` Z )
25		pwssplit1.f	. . . 4 |- F = ( x e. B |-> ( x |` V ) )
26	9, 14, 1, 24, 25	pwssplit2 19060	. . 3 |- ( ( W e. Grp /\ U e. X /\ V C_ U ) -> F e. ( Y GrpHom Z ) )
27	23, 26	syl3an1 1359	. 2 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> F e. ( Y GrpHom Z ) )
28		snex 4908	. . . . . . . 8 |- { a } e. _V
29		xpexg 6960	. . . . . . . 8 |- ( ( U e. X /\ { a } e. _V ) -> ( U X. { a } ) e. _V )
30	8, 28, 29	sylancl 694	. . . . . . 7 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> ( U X. { a } ) e. _V )
31		vex 3203	. . . . . . 7 |- b e. _V
32		offres 7163	. . . . . . 7 |- ( ( ( U X. { a } ) e. _V /\ b e. _V ) -> ( ( ( U X. { a } ) oF ( .s ` W ) b ) |` V ) = ( ( ( U X. { a } ) |` V ) oF ( .s ` W ) ( b |` V ) ) )
33	30, 31, 32	sylancl 694	. . . . . 6 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> ( ( ( U X. { a } ) oF ( .s ` W ) b ) |` V ) = ( ( ( U X. { a } ) |` V ) oF ( .s ` W ) ( b |` V ) ) )
34	33	adantr 481	. . . . 5 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( ( U X. { a } ) oF ( .s ` W ) b ) |` V ) = ( ( ( U X. { a } ) |` V ) oF ( .s ` W ) ( b |` V ) ) )
35		xpssres 5434	. . . . . . . 8 |- ( V C_ U -> ( ( U X. { a } ) |` V ) = ( V X. { a } ) )
36	35	3ad2ant3 1084	. . . . . . 7 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> ( ( U X. { a } ) |` V ) = ( V X. { a } ) )
37	36	adantr 481	. . . . . 6 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( U X. { a } ) |` V ) = ( V X. { a } ) )
38	37	oveq1d 6665	. . . . 5 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( ( U X. { a } ) |` V ) oF ( .s ` W ) ( b |` V ) ) = ( ( V X. { a } ) oF ( .s ` W ) ( b |` V ) ) )
39	34, 38	eqtrd 2656	. . . 4 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( ( U X. { a } ) oF ( .s ` W ) b ) |` V ) = ( ( V X. { a } ) oF ( .s ` W ) ( b |` V ) ) )
40		eqid 2622	. . . . . 6 |- ( .s ` W ) = ( .s ` W )
41		eqid 2622	. . . . . 6 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
42		simpl1 1064	. . . . . 6 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> W e. LMod )
43		simpl2 1065	. . . . . 6 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> U e. X )
44	21	fveq2d 6195	. . . . . . . . 9 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` Y ) ) )
45	44	eleq2d 2687	. . . . . . . 8 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> ( a e. ( Base ` (Scalar ` W ) ) <-> a e. ( Base ` (Scalar ` Y ) ) ) )
46	45	biimpar 502	. . . . . . 7 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ a e. ( Base ` (Scalar ` Y ) ) ) -> a e. ( Base ` (Scalar ` W ) ) )
47	46	adantrr 753	. . . . . 6 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> a e. ( Base ` (Scalar ` W ) ) )
48		simprr 796	. . . . . 6 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> b e. B )
49	9, 1, 40, 2, 17, 41, 42, 43, 47, 48	pwsvscafval 16154	. . . . 5 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( a ( .s ` Y ) b ) = ( ( U X. { a } ) oF ( .s ` W ) b ) )
50	49	reseq1d 5395	. . . 4 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( a ( .s ` Y ) b ) |` V ) = ( ( ( U X. { a } ) oF ( .s ` W ) b ) |` V ) )
51	25	fvtresfn 6284	. . . . . 6 |- ( b e. B -> ( F ` b ) = ( b |` V ) )
52	51	ad2antll 765	. . . . 5 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( F ` b ) = ( b |` V ) )
53	52	oveq2d 6666	. . . 4 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( V X. { a } ) oF ( .s ` W ) ( F ` b ) ) = ( ( V X. { a } ) oF ( .s ` W ) ( b |` V ) ) )
54	39, 50, 53	3eqtr4d 2666	. . 3 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( ( a ( .s ` Y ) b ) |` V ) = ( ( V X. { a } ) oF ( .s ` W ) ( F ` b ) ) )
55	1, 4, 2, 6	lmodvscl 18880	. . . . . 6 |- ( ( Y e. LMod /\ a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) -> ( a ( .s ` Y ) b ) e. B )
56	55	3expb 1266	. . . . 5 |- ( ( Y e. LMod /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( a ( .s ` Y ) b ) e. B )
57	11, 56	sylan 488	. . . 4 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( a ( .s ` Y ) b ) e. B )
58	25	fvtresfn 6284	. . . 4 |- ( ( a ( .s ` Y ) b ) e. B -> ( F ` ( a ( .s ` Y ) b ) ) = ( ( a ( .s ` Y ) b ) |` V ) )
59	57, 58	syl 17	. . 3 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( F ` ( a ( .s ` Y ) b ) ) = ( ( a ( .s ` Y ) b ) |` V ) )
60	13	adantr 481	. . . 4 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> V e. _V )
61	9, 14, 1, 24, 25	pwssplit0 19058	. . . . . 6 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> F : B --> C )
62	61	ffvelrnda 6359	. . . . 5 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ b e. B ) -> ( F ` b ) e. C )
63	62	adantrl 752	. . . 4 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( F ` b ) e. C )
64	14, 24, 40, 3, 17, 41, 42, 60, 47, 63	pwsvscafval 16154	. . 3 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( a ( .s ` Z ) ( F ` b ) ) = ( ( V X. { a } ) oF ( .s ` W ) ( F ` b ) ) )
65	54, 59, 64	3eqtr4d 2666	. 2 |- ( ( ( W e. LMod /\ U e. X /\ V C_ U ) /\ ( a e. ( Base ` (Scalar ` Y ) ) /\ b e. B ) ) -> ( F ` ( a ( .s ` Y ) b ) ) = ( a ( .s ` Z ) ( F ` b ) ) )
66	1, 2, 3, 4, 5, 6, 11, 16, 22, 27, 65	islmhmd 19039	1 |- ( ( W e. LMod /\ U e. X /\ V C_ U ) -> F e. ( Y LMHom Z ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   |-> cmpt 4729   X. cxp 5112   |` cres 5116   ` cfv 5888  (class class class)co 6650   oFcof 6895   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   ^_s cpws 16107   Grpcgrp 17422   GrpHom cghm 17657   LModclmod 18863   LMHom clmhm 19019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-prds 16108  df-pws 16110  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-lmod 18865  df-lmhm 19022

This theorem is referenced by:  frlmsplit2  20112  pwssplit4  37659  pwslnmlem2  37663