Theorem pj1lmhm 18323

Description: The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
pj1lmhm.l	|- L = ( LSubSp ` W )
pj1lmhm.s	|- .(+) = ( LSSum ` W )
pj1lmhm.z	|- .0. = ( 0g ` W )
pj1lmhm.p	|- P = ( proj1 ` W )
pj1lmhm.1	|- ( ph -> W e. LMod )
pj1lmhm.2	|- ( ph -> T e. L )
pj1lmhm.3	|- ( ph -> U e. L )
pj1lmhm.4	|- ( ph -> ( T i^i U ) = { .0. } )

Assertion

Ref	Expression
pj1lmhm	|- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) )

Proof of Theorem pj1lmhm

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

Step	Hyp	Ref	Expression
1		eqid 2451	. . 3 |- ( +g ` W ) = ( +g ` W )
2		pj1lmhm.s	. . 3 |- .(+) = ( LSSum ` W )
3		pj1lmhm.z	. . 3 |- .0. = ( 0g ` W )
4		eqid 2451	. . 3 |- (Cntz ` W ) = (Cntz ` W )
5		pj1lmhm.1	. . . . 5 |- ( ph -> W e. LMod )
6		pj1lmhm.l	. . . . . 6 |- L = ( LSubSp ` W )
7	6	lsssssubg 18181	. . . . 5 |- ( W e. LMod -> L C_ (SubGrp ` W ) )
8	5, 7	syl 17	. . . 4 |- ( ph -> L C_ (SubGrp ` W ) )
9		pj1lmhm.2	. . . 4 |- ( ph -> T e. L )
10	8, 9	sseldd 3433	. . 3 |- ( ph -> T e. (SubGrp ` W ) )
11		pj1lmhm.3	. . . 4 |- ( ph -> U e. L )
12	8, 11	sseldd 3433	. . 3 |- ( ph -> U e. (SubGrp ` W ) )
13		pj1lmhm.4	. . 3 |- ( ph -> ( T i^i U ) = { .0. } )
14		lmodabl 18135	. . . . 5 |- ( W e. LMod -> W e. Abel )
15	5, 14	syl 17	. . . 4 |- ( ph -> W e. Abel )
16	4, 15, 10, 12	ablcntzd 17495	. . 3 |- ( ph -> T C_ ( (Cntz ` W ) ` U ) )
17		pj1lmhm.p	. . 3 |- P = ( proj1 ` W )
18	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1ghm 17353	. 2 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) GrpHom W ) )
19		eqid 2451	. . 3 |- (Scalar ` W ) = (Scalar ` W )
20	19	a1i 11	. 2 |- ( ph -> (Scalar ` W ) = (Scalar ` W ) )
21	1, 2, 3, 4, 10, 12, 13, 16, 17	pj1id 17349	. . . . . . . . 9 |- ( ( ph /\ y e. ( T .(+) U ) ) -> y = ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) )
22	21	adantrl 722	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> y = ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) )
23	22	oveq2d 6306	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) = ( x ( .s ` W ) ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) ) )
24	5	adantr 467	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> W e. LMod )
25		simprl 764	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> x e. ( Base ` (Scalar ` W ) ) )
26	9	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T e. L )
27		eqid 2451	. . . . . . . . . . 11 |- ( Base ` W ) = ( Base ` W )
28	27, 6	lssss 18160	. . . . . . . . . 10 |- ( T e. L -> T C_ ( Base ` W ) )
29	26, 28	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T C_ ( Base ` W ) )
30	10	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T e. (SubGrp ` W ) )
31	12	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U e. (SubGrp ` W ) )
32	13	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T i^i U ) = { .0. } )
33	16	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T C_ ( (Cntz ` W ) ` U ) )
34	1, 2, 3, 4, 30, 31, 32, 33, 17	pj1f 17347	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T P U ) : ( T .(+) U ) --> T )
35		simprr 766	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> y e. ( T .(+) U ) )
36	34, 35	ffvelrnd 6023	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. T )
37	29, 36	sseldd 3433	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. ( Base ` W ) )
38	11	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U e. L )
39	27, 6	lssss 18160	. . . . . . . . . 10 |- ( U e. L -> U C_ ( Base ` W ) )
40	38, 39	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U C_ ( Base ` W ) )
41	1, 2, 3, 4, 30, 31, 32, 33, 17	pj2f 17348	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( U P T ) : ( T .(+) U ) --> U )
42	41, 35	ffvelrnd 6023	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. U )
43	40, 42	sseldd 3433	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. ( Base ` W ) )
44		eqid 2451	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
45		eqid 2451	. . . . . . . . 9 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
46	27, 1, 19, 44, 45	lmodvsdi 18114	. . . . . . . 8 |- ( ( W e. LMod /\ ( x e. ( Base ` (Scalar ` W ) ) /\ ( ( T P U ) ` y ) e. ( Base ` W ) /\ ( ( U P T ) ` y ) e. ( Base ` W ) ) ) -> ( x ( .s ` W ) ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) ) = ( ( x ( .s ` W ) ( ( T P U ) ` y ) ) ( +g ` W ) ( x ( .s ` W ) ( ( U P T ) ` y ) ) ) )
47	24, 25, 37, 43, 46	syl13anc 1270	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) ) = ( ( x ( .s ` W ) ( ( T P U ) ` y ) ) ( +g ` W ) ( x ( .s ` W ) ( ( U P T ) ` y ) ) ) )
48	23, 47	eqtrd 2485	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) = ( ( x ( .s ` W ) ( ( T P U ) ` y ) ) ( +g ` W ) ( x ( .s ` W ) ( ( U P T ) ` y ) ) ) )
49	6, 2	lsmcl 18306	. . . . . . . . . 10 |- ( ( W e. LMod /\ T e. L /\ U e. L ) -> ( T .(+) U ) e. L )
50	5, 9, 11, 49	syl3anc 1268	. . . . . . . . 9 |- ( ph -> ( T .(+) U ) e. L )
51	50	adantr 467	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T .(+) U ) e. L )
52	19, 44, 45, 6	lssvscl 18178	. . . . . . . 8 |- ( ( ( W e. LMod /\ ( T .(+) U ) e. L ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) e. ( T .(+) U ) )
53	24, 51, 25, 35, 52	syl22anc 1269	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) e. ( T .(+) U ) )
54	19, 44, 45, 6	lssvscl 18178	. . . . . . . 8 |- ( ( ( W e. LMod /\ T e. L ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ ( ( T P U ) ` y ) e. T ) ) -> ( x ( .s ` W ) ( ( T P U ) ` y ) ) e. T )
55	24, 26, 25, 36, 54	syl22anc 1269	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) ( ( T P U ) ` y ) ) e. T )
56	19, 44, 45, 6	lssvscl 18178	. . . . . . . 8 |- ( ( ( W e. LMod /\ U e. L ) /\ ( x e. ( Base ` (Scalar ` W ) ) /\ ( ( U P T ) ` y ) e. U ) ) -> ( x ( .s ` W ) ( ( U P T ) ` y ) ) e. U )
57	24, 38, 25, 42, 56	syl22anc 1269	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) ( ( U P T ) ` y ) ) e. U )
58	1, 2, 3, 4, 30, 31, 32, 33, 17, 53, 55, 57	pj1eq 17350	. . . . . 6 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( x ( .s ` W ) y ) = ( ( x ( .s ` W ) ( ( T P U ) ` y ) ) ( +g ` W ) ( x ( .s ` W ) ( ( U P T ) ` y ) ) ) <-> ( ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) /\ ( ( U P T ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( U P T ) ` y ) ) ) ) )
59	48, 58	mpbid 214	. . . . 5 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) /\ ( ( U P T ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( U P T ) ` y ) ) ) )
60	59	simpld 461	. . . 4 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) )
61	60	ralrimivva 2809	. . 3 |- ( ph -> A. x e. ( Base ` (Scalar ` W ) ) A. y e. ( T .(+) U ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) )
62	8, 50	sseldd 3433	. . . . . 6 |- ( ph -> ( T .(+) U ) e. (SubGrp ` W ) )
63		eqid 2451	. . . . . . 7 |- ( W ↾s ( T .(+) U ) ) = ( W ↾s ( T .(+) U ) )
64	63	subgbas 16821	. . . . . 6 |- ( ( T .(+) U ) e. (SubGrp ` W ) -> ( T .(+) U ) = ( Base ` ( W ↾s ( T .(+) U ) ) ) )
65	62, 64	syl 17	. . . . 5 |- ( ph -> ( T .(+) U ) = ( Base ` ( W ↾s ( T .(+) U ) ) ) )
66	65	raleqdv 2993	. . . 4 |- ( ph -> ( A. y e. ( T .(+) U ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) <-> A. y e. ( Base ` ( W ↾s ( T .(+) U ) ) ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) ) )
67	66	ralbidv 2827	. . 3 |- ( ph -> ( A. x e. ( Base ` (Scalar ` W ) ) A. y e. ( T .(+) U ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) <-> A. x e. ( Base ` (Scalar ` W ) ) A. y e. ( Base ` ( W ↾s ( T .(+) U ) ) ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) ) )
68	61, 67	mpbid 214	. 2 |- ( ph -> A. x e. ( Base ` (Scalar ` W ) ) A. y e. ( Base ` ( W ↾s ( T .(+) U ) ) ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) )
69	63, 6	lsslmod 18183	. . . 4 |- ( ( W e. LMod /\ ( T .(+) U ) e. L ) -> ( W ↾s ( T .(+) U ) ) e. LMod )
70	5, 50, 69	syl2anc 667	. . 3 |- ( ph -> ( W ↾s ( T .(+) U ) ) e. LMod )
71		ovex 6318	. . . . 5 |- ( T .(+) U ) e. _V
72	63, 19	resssca 15275	. . . . 5 |- ( ( T .(+) U ) e. _V -> (Scalar ` W ) = (Scalar ` ( W ↾s ( T .(+) U ) ) ) )
73	71, 72	ax-mp 5	. . . 4 |- (Scalar ` W ) = (Scalar ` ( W ↾s ( T .(+) U ) ) )
74		eqid 2451	. . . 4 |- ( Base ` ( W ↾s ( T .(+) U ) ) ) = ( Base ` ( W ↾s ( T .(+) U ) ) )
75	63, 44	ressvsca 15276	. . . . 5 |- ( ( T .(+) U ) e. _V -> ( .s ` W ) = ( .s ` ( W ↾s ( T .(+) U ) ) ) )
76	71, 75	ax-mp 5	. . . 4 |- ( .s ` W ) = ( .s ` ( W ↾s ( T .(+) U ) ) )
77	73, 19, 45, 74, 76, 44	islmhm3 18251	. . 3 |- ( ( ( W ↾s ( T .(+) U ) ) e. LMod /\ W e. LMod ) -> ( ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) <-> ( ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) GrpHom W ) /\ (Scalar ` W ) = (Scalar ` W ) /\ A. x e. ( Base ` (Scalar ` W ) ) A. y e. ( Base ` ( W ↾s ( T .(+) U ) ) ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) ) ) )
78	70, 5, 77	syl2anc 667	. 2 |- ( ph -> ( ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) <-> ( ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) GrpHom W ) /\ (Scalar ` W ) = (Scalar ` W ) /\ A. x e. ( Base ` (Scalar ` W ) ) A. y e. ( Base ` ( W ↾s ( T .(+) U ) ) ) ( ( T P U ) ` ( x ( .s ` W ) y ) ) = ( x ( .s ` W ) ( ( T P U ) ` y ) ) ) ) )
79	18, 20, 68, 78	mpbir3and 1191	1 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   _Vcvv 3045   i^i cin 3403   C_ wss 3404   {csn 3968   ` cfv 5582  (class class class)co 6290   Basecbs 15121   ↾_s cress 15122   +g cplusg 15190  Scalarcsca 15193   .scvsca 15194   0gc0g 15338  SubGrpcsubg 16811   GrpHom cghm 16880  Cntzccntz 16969   LSSumclsm 17286   proj1cpj1 17287   Abelcabl 17431   LModclmod 18091   LSubSpclss 18155   LMHom clmhm 18242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-ndx 15124  df-slot 15125  df-base 15126  df-sets 15127  df-ress 15128  df-plusg 15203  df-sca 15206  df-vsca 15207  df-0g 15340  df-mgm 16488  df-sgrp 16527  df-mnd 16537  df-submnd 16583  df-grp 16673  df-minusg 16674  df-sbg 16675  df-subg 16814  df-ghm 16881  df-cntz 16971  df-lsm 17288  df-pj1 17289  df-cmn 17432  df-abl 17433  df-mgp 17724  df-ur 17736  df-ring 17782  df-lmod 18093  df-lss 18156  df-lmhm 18245

This theorem is referenced by:  pj1lmhm2  18324  pjff  19275