Theorem frlmsplit2 20112

Description: Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)

Hypotheses

Ref	Expression
frlmsplit2.y	|- Y = ( R freeLMod U )
frlmsplit2.z	|- Z = ( R freeLMod V )
frlmsplit2.b	|- B = ( Base ` Y )
frlmsplit2.c	|- C = ( Base ` Z )
frlmsplit2.f	|- F = ( x e. B |-> ( x |` V ) )

Assertion

Ref	Expression
frlmsplit2	|- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> F e. ( Y LMHom Z ) )

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

Allowed substitution hint:   F( x)

Proof of Theorem frlmsplit2

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . . 6 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> R e. Ring )
2		simp2 1062	. . . . . 6 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> U e. X )
3		frlmsplit2.y	. . . . . . 7 |- Y = ( R freeLMod U )
4		frlmsplit2.b	. . . . . . 7 |- B = ( Base ` Y )
5		eqid 2622	. . . . . . 7 |- ( LSubSp ` ( (ringLMod ` R ) ^s U ) ) = ( LSubSp ` ( (ringLMod ` R ) ^s U ) )
6	3, 4, 5	frlmlss 20095	. . . . . 6 |- ( ( R e. Ring /\ U e. X ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s U ) ) )
7	1, 2, 6	syl2anc 693	. . . . 5 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> B e. ( LSubSp ` ( (ringLMod ` R ) ^s U ) ) )
8		eqid 2622	. . . . . 6 |- ( Base ` ( (ringLMod ` R ) ^s U ) ) = ( Base ` ( (ringLMod ` R ) ^s U ) )
9	8, 5	lssss 18937	. . . . 5 |- ( B e. ( LSubSp ` ( (ringLMod ` R ) ^s U ) ) -> B C_ ( Base ` ( (ringLMod ` R ) ^s U ) ) )
10		resmpt 5449	. . . . 5 |- ( B C_ ( Base ` ( (ringLMod ` R ) ^s U ) ) -> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) = ( x e. B |-> ( x |` V ) ) )
11	7, 9, 10	3syl 18	. . . 4 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) = ( x e. B |-> ( x |` V ) ) )
12		frlmsplit2.f	. . . 4 |- F = ( x e. B |-> ( x |` V ) )
13	11, 12	syl6eqr 2674	. . 3 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) = F )
14		rlmlmod 19205	. . . . . 6 |- ( R e. Ring -> (ringLMod ` R ) e. LMod )
15		eqid 2622	. . . . . . 7 |- ( (ringLMod ` R ) ^s U ) = ( (ringLMod ` R ) ^s U )
16		eqid 2622	. . . . . . 7 |- ( (ringLMod ` R ) ^s V ) = ( (ringLMod ` R ) ^s V )
17		eqid 2622	. . . . . . 7 |- ( Base ` ( (ringLMod ` R ) ^s V ) ) = ( Base ` ( (ringLMod ` R ) ^s V ) )
18		eqid 2622	. . . . . . 7 |- ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) = ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) )
19	15, 16, 8, 17, 18	pwssplit3 19061	. . . . . 6 |- ( ( (ringLMod ` R ) e. LMod /\ U e. X /\ V C_ U ) -> ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) e. ( ( (ringLMod ` R ) ^s U ) LMHom ( (ringLMod ` R ) ^s V ) ) )
20	14, 19	syl3an1 1359	. . . . 5 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) e. ( ( (ringLMod ` R ) ^s U ) LMHom ( (ringLMod ` R ) ^s V ) ) )
21		eqid 2622	. . . . . 6 |- ( ( (ringLMod ` R ) ^s U ) ↾s B ) = ( ( (ringLMod ` R ) ^s U ) ↾s B )
22	5, 21	reslmhm 19052	. . . . 5 |- ( ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) e. ( ( (ringLMod ` R ) ^s U ) LMHom ( (ringLMod ` R ) ^s V ) ) /\ B e. ( LSubSp ` ( (ringLMod ` R ) ^s U ) ) ) -> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( (ringLMod ` R ) ^s V ) ) )
23	20, 7, 22	syl2anc 693	. . . 4 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( (ringLMod ` R ) ^s V ) ) )
24	14	3ad2ant1 1082	. . . . . 6 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> (ringLMod ` R ) e. LMod )
25		simp3 1063	. . . . . . 7 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> V C_ U )
26	2, 25	ssexd 4805	. . . . . 6 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> V e. _V )
27	16	pwslmod 18970	. . . . . 6 |- ( ( (ringLMod ` R ) e. LMod /\ V e. _V ) -> ( (ringLMod ` R ) ^s V ) e. LMod )
28	24, 26, 27	syl2anc 693	. . . . 5 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( (ringLMod ` R ) ^s V ) e. LMod )
29		frlmsplit2.z	. . . . . . 7 |- Z = ( R freeLMod V )
30		frlmsplit2.c	. . . . . . 7 |- C = ( Base ` Z )
31		eqid 2622	. . . . . . 7 |- ( LSubSp ` ( (ringLMod ` R ) ^s V ) ) = ( LSubSp ` ( (ringLMod ` R ) ^s V ) )
32	29, 30, 31	frlmlss 20095	. . . . . 6 |- ( ( R e. Ring /\ V e. _V ) -> C e. ( LSubSp ` ( (ringLMod ` R ) ^s V ) ) )
33	1, 26, 32	syl2anc 693	. . . . 5 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> C e. ( LSubSp ` ( (ringLMod ` R ) ^s V ) ) )
34	11	rneqd 5353	. . . . . 6 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ran ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) = ran ( x e. B |-> ( x |` V ) ) )
35		eqid 2622	. . . . . . . . . . . . 13 |- ( Base ` R ) = ( Base ` R )
36	3, 35, 4	frlmbasf 20104	. . . . . . . . . . . 12 |- ( ( U e. X /\ x e. B ) -> x : U --> ( Base ` R ) )
37	2, 36	sylan 488	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> x : U --> ( Base ` R ) )
38		simpl3 1066	. . . . . . . . . . 11 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> V C_ U )
39	37, 38	fssresd 6071	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( x |` V ) : V --> ( Base ` R ) )
40		fvex 6201	. . . . . . . . . . . 12 |- ( Base ` R ) e. _V
41		elmapg 7870	. . . . . . . . . . . 12 |- ( ( ( Base ` R ) e. _V /\ V e. _V ) -> ( ( x |` V ) e. ( ( Base ` R ) ^m V ) <-> ( x |` V ) : V --> ( Base ` R ) ) )
42	40, 26, 41	sylancr 695	. . . . . . . . . . 11 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( x |` V ) e. ( ( Base ` R ) ^m V ) <-> ( x |` V ) : V --> ( Base ` R ) ) )
43	42	adantr 481	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( ( x |` V ) e. ( ( Base ` R ) ^m V ) <-> ( x |` V ) : V --> ( Base ` R ) ) )
44	39, 43	mpbird 247	. . . . . . . . 9 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( x |` V ) e. ( ( Base ` R ) ^m V ) )
45		eqid 2622	. . . . . . . . . . . 12 |- ( 0g ` R ) = ( 0g ` R )
46	3, 45, 4	frlmbasfsupp 20102	. . . . . . . . . . 11 |- ( ( U e. X /\ x e. B ) -> x finSupp ( 0g ` R ) )
47	2, 46	sylan 488	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> x finSupp ( 0g ` R ) )
48		fvexd 6203	. . . . . . . . . 10 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( 0g ` R ) e. _V )
49	47, 48	fsuppres 8300	. . . . . . . . 9 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( x |` V ) finSupp ( 0g ` R ) )
50	29, 35, 45, 30	frlmelbas 20100	. . . . . . . . . . 11 |- ( ( R e. Ring /\ V e. _V ) -> ( ( x |` V ) e. C <-> ( ( x |` V ) e. ( ( Base ` R ) ^m V ) /\ ( x |` V ) finSupp ( 0g ` R ) ) ) )
51	1, 26, 50	syl2anc 693	. . . . . . . . . 10 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( x |` V ) e. C <-> ( ( x |` V ) e. ( ( Base ` R ) ^m V ) /\ ( x |` V ) finSupp ( 0g ` R ) ) ) )
52	51	adantr 481	. . . . . . . . 9 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( ( x |` V ) e. C <-> ( ( x |` V ) e. ( ( Base ` R ) ^m V ) /\ ( x |` V ) finSupp ( 0g ` R ) ) ) )
53	44, 49, 52	mpbir2and 957	. . . . . . . 8 |- ( ( ( R e. Ring /\ U e. X /\ V C_ U ) /\ x e. B ) -> ( x |` V ) e. C )
54		eqid 2622	. . . . . . . 8 |- ( x e. B |-> ( x |` V ) ) = ( x e. B |-> ( x |` V ) )
55	53, 54	fmptd 6385	. . . . . . 7 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( x e. B |-> ( x |` V ) ) : B --> C )
56		frn 6053	. . . . . . 7 |- ( ( x e. B |-> ( x |` V ) ) : B --> C -> ran ( x e. B |-> ( x |` V ) ) C_ C )
57	55, 56	syl 17	. . . . . 6 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ran ( x e. B |-> ( x |` V ) ) C_ C )
58	34, 57	eqsstrd 3639	. . . . 5 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ran ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) C_ C )
59		eqid 2622	. . . . . 6 |- ( ( (ringLMod ` R ) ^s V ) ↾s C ) = ( ( (ringLMod ` R ) ^s V ) ↾s C )
60	59, 31	reslmhm2b 19054	. . . . 5 |- ( ( ( (ringLMod ` R ) ^s V ) e. LMod /\ C e. ( LSubSp ` ( (ringLMod ` R ) ^s V ) ) /\ ran ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) C_ C ) -> ( ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( (ringLMod ` R ) ^s V ) ) <-> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( ( (ringLMod ` R ) ^s V ) ↾s C ) ) ) )
61	28, 33, 58, 60	syl3anc 1326	. . . 4 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( (ringLMod ` R ) ^s V ) ) <-> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( ( (ringLMod ` R ) ^s V ) ↾s C ) ) ) )
62	23, 61	mpbid 222	. . 3 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( ( x e. ( Base ` ( (ringLMod ` R ) ^s U ) ) |-> ( x |` V ) ) |` B ) e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( ( (ringLMod ` R ) ^s V ) ↾s C ) ) )
63	13, 62	eqeltrrd 2702	. 2 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> F e. ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( ( (ringLMod ` R ) ^s V ) ↾s C ) ) )
64	3, 4	frlmpws 20094	. . . 4 |- ( ( R e. Ring /\ U e. X ) -> Y = ( ( (ringLMod ` R ) ^s U ) ↾s B ) )
65	1, 2, 64	syl2anc 693	. . 3 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> Y = ( ( (ringLMod ` R ) ^s U ) ↾s B ) )
66	29, 30	frlmpws 20094	. . . 4 |- ( ( R e. Ring /\ V e. _V ) -> Z = ( ( (ringLMod ` R ) ^s V ) ↾s C ) )
67	1, 26, 66	syl2anc 693	. . 3 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> Z = ( ( (ringLMod ` R ) ^s V ) ↾s C ) )
68	65, 67	oveq12d 6668	. 2 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> ( Y LMHom Z ) = ( ( ( (ringLMod ` R ) ^s U ) ↾s B ) LMHom ( ( (ringLMod ` R ) ^s V ) ↾s C ) ) )
69	63, 68	eleqtrrd 2704	1 |- ( ( R e. Ring /\ U e. X /\ V C_ U ) -> F e. ( Y LMHom Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   finSupp cfsupp 8275   Basecbs 15857   ↾_s cress 15858   0gc0g 16100   ^_s cpws 16107   Ringcrg 18547   LModclmod 18863   LSubSpclss 18932   LMHom clmhm 19019  ringLModcrglmod 19169   freeLMod cfrlm 20090

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-supp 7296  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-fsupp 8276  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-ress 15865  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-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lmhm 19022  df-sra 19172  df-rgmod 19173  df-dsmm 20076  df-frlm 20091

This theorem is referenced by:  frlmsslss  20113