Theorem pj1lmhm 17181

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 2443	. . 3 |- ( +g ` W ) = ( +g ` W )
2		pj1lmhm.s	. . 3 |- .(+) = ( LSSum ` W )
3		pj1lmhm.z	. . 3 |- .0. = ( 0g ` W )
4		eqid 2443	. . 3 |- (Cntz ` W ) = (Cntz ` W )
5		pj1lmhm.1	. . . . 5 |- ( ph -> W e. LMod )
6		pj1lmhm.l	. . . . . 6 |- L = ( LSubSp ` W )
7	6	lsssssubg 17039	. . . . 5 |- ( W e. LMod -> L C_ (SubGrp ` W ) )
8	5, 7	syl 16	. . . 4 |- ( ph -> L C_ (SubGrp ` W ) )
9		pj1lmhm.2	. . . 4 |- ( ph -> T e. L )
10	8, 9	sseldd 3357	. . 3 |- ( ph -> T e. (SubGrp ` W ) )
11		pj1lmhm.3	. . . 4 |- ( ph -> U e. L )
12	8, 11	sseldd 3357	. . 3 |- ( ph -> U e. (SubGrp ` W ) )
13		pj1lmhm.4	. . 3 |- ( ph -> ( T i^i U ) = { .0. } )
14		lmodabl 16992	. . . . 5 |- ( W e. LMod -> W e. Abel )
15	5, 14	syl 16	. . . 4 |- ( ph -> W e. Abel )
16	4, 15, 10, 12	ablcntzd 16339	. . 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 16200	. 2 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) GrpHom W ) )
19		eqid 2443	. . 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 16196	. . . . . . . . 9 |- ( ( ph /\ y e. ( T .(+) U ) ) -> y = ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) )
22	21	adantrl 715	. . . . . . . 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 6107	. . . . . . 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 465	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> W e. LMod )
25		simprl 755	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> x e. ( Base ` (Scalar ` W ) ) )
26	9	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T e. L )
27		eqid 2443	. . . . . . . . . . 11 |- ( Base ` W ) = ( Base ` W )
28	27, 6	lssss 17018	. . . . . . . . . 10 |- ( T e. L -> T C_ ( Base ` W ) )
29	26, 28	syl 16	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T C_ ( Base ` W ) )
30	10	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T e. (SubGrp ` W ) )
31	12	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U e. (SubGrp ` W ) )
32	13	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T i^i U ) = { .0. } )
33	16	adantr 465	. . . . . . . . . . 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 16194	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T P U ) : ( T .(+) U ) --> T )
35		simprr 756	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> y e. ( T .(+) U ) )
36	34, 35	ffvelrnd 5844	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. T )
37	29, 36	sseldd 3357	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. ( Base ` W ) )
38	11	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U e. L )
39	27, 6	lssss 17018	. . . . . . . . . 10 |- ( U e. L -> U C_ ( Base ` W ) )
40	38, 39	syl 16	. . . . . . . . 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 16195	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( U P T ) : ( T .(+) U ) --> U )
42	41, 35	ffvelrnd 5844	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. U )
43	40, 42	sseldd 3357	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. ( Base ` W ) )
44		eqid 2443	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
45		eqid 2443	. . . . . . . . 9 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
46	27, 1, 19, 44, 45	lmodvsdi 16971	. . . . . . . 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 1220	. . . . . . 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 2475	. . . . . 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 17164	. . . . . . . . . 10 |- ( ( W e. LMod /\ T e. L /\ U e. L ) -> ( T .(+) U ) e. L )
50	5, 9, 11, 49	syl3anc 1218	. . . . . . . . 9 |- ( ph -> ( T .(+) U ) e. L )
51	50	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T .(+) U ) e. L )
52	19, 44, 45, 6	lssvscl 17036	. . . . . . . 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 1219	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) e. ( T .(+) U ) )
54	19, 44, 45, 6	lssvscl 17036	. . . . . . . 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 1219	. . . . . . 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 17036	. . . . . . . 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 1219	. . . . . . 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 16197	. . . . . 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 210	. . . . 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 459	. . . 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 2808	. . 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 3357	. . . . . 6 |- ( ph -> ( T .(+) U ) e. (SubGrp ` W ) )
63		eqid 2443	. . . . . . 7 |- ( W ↾s ( T .(+) U ) ) = ( W ↾s ( T .(+) U ) )
64	63	subgbas 15685	. . . . . 6 |- ( ( T .(+) U ) e. (SubGrp ` W ) -> ( T .(+) U ) = ( Base ` ( W ↾s ( T .(+) U ) ) ) )
65	62, 64	syl 16	. . . . 5 |- ( ph -> ( T .(+) U ) = ( Base ` ( W ↾s ( T .(+) U ) ) ) )
66	65	raleqdv 2923	. . . 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 2735	. . 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 210	. 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 17041	. . . 4 |- ( ( W e. LMod /\ ( T .(+) U ) e. L ) -> ( W ↾s ( T .(+) U ) ) e. LMod )
70	5, 50, 69	syl2anc 661	. . 3 |- ( ph -> ( W ↾s ( T .(+) U ) ) e. LMod )
71		ovex 6116	. . . . 5 |- ( T .(+) U ) e. _V
72	63, 19	resssca 14316	. . . . 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 2443	. . . 4 |- ( Base ` ( W ↾s ( T .(+) U ) ) ) = ( Base ` ( W ↾s ( T .(+) U ) ) )
75	63, 44	ressvsca 14317	. . . . 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 17109	. . 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 661	. 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 1171	1 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   A.wral 2715   _Vcvv 2972   i^i cin 3327   C_ wss 3328   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174   ↾_s cress 14175   +g cplusg 14238  Scalarcsca 14241   .scvsca 14242   0gc0g 14378  SubGrpcsubg 15675   GrpHom cghm 15744  Cntzccntz 15833   LSSumclsm 16133   proj1cpj1 16134   Abelcabel 16278   LModclmod 16948   LSubSpclss 17013   LMHom clmhm 17100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-sca 14254  df-vsca 14255  df-0g 14380  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-ghm 15745  df-cntz 15835  df-lsm 16135  df-pj1 16136  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-lmod 16950  df-lss 17014  df-lmhm 17103

This theorem is referenced by:  pj1lmhm2  17182  pjff  18137