Theorem pj1lmhm 17958

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 2402	. . 3 |- ( +g ` W ) = ( +g ` W )
2		pj1lmhm.s	. . 3 |- .(+) = ( LSSum ` W )
3		pj1lmhm.z	. . 3 |- .0. = ( 0g ` W )
4		eqid 2402	. . 3 |- (Cntz ` W ) = (Cntz ` W )
5		pj1lmhm.1	. . . . 5 |- ( ph -> W e. LMod )
6		pj1lmhm.l	. . . . . 6 |- L = ( LSubSp ` W )
7	6	lsssssubg 17816	. . . . 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 3442	. . 3 |- ( ph -> T e. (SubGrp ` W ) )
11		pj1lmhm.3	. . . 4 |- ( ph -> U e. L )
12	8, 11	sseldd 3442	. . 3 |- ( ph -> U e. (SubGrp ` W ) )
13		pj1lmhm.4	. . 3 |- ( ph -> ( T i^i U ) = { .0. } )
14		lmodabl 17769	. . . . 5 |- ( W e. LMod -> W e. Abel )
15	5, 14	syl 17	. . . 4 |- ( ph -> W e. Abel )
16	4, 15, 10, 12	ablcntzd 17079	. . 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 16937	. 2 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) GrpHom W ) )
19		eqid 2402	. . 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 16933	. . . . . . . . 9 |- ( ( ph /\ y e. ( T .(+) U ) ) -> y = ( ( ( T P U ) ` y ) ( +g ` W ) ( ( U P T ) ` y ) ) )
22	21	adantrl 714	. . . . . . . 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 6250	. . . . . . 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 463	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> W e. LMod )
25		simprl 756	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> x e. ( Base ` (Scalar ` W ) ) )
26	9	adantr 463	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T e. L )
27		eqid 2402	. . . . . . . . . . 11 |- ( Base ` W ) = ( Base ` W )
28	27, 6	lssss 17795	. . . . . . . . . 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 463	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> T e. (SubGrp ` W ) )
31	12	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U e. (SubGrp ` W ) )
32	13	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T i^i U ) = { .0. } )
33	16	adantr 463	. . . . . . . . . . 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 16931	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T P U ) : ( T .(+) U ) --> T )
35		simprr 758	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> y e. ( T .(+) U ) )
36	34, 35	ffvelrnd 5966	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. T )
37	29, 36	sseldd 3442	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. ( Base ` W ) )
38	11	adantr 463	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> U e. L )
39	27, 6	lssss 17795	. . . . . . . . . 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 16932	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( U P T ) : ( T .(+) U ) --> U )
42	41, 35	ffvelrnd 5966	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. U )
43	40, 42	sseldd 3442	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. ( Base ` W ) )
44		eqid 2402	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
45		eqid 2402	. . . . . . . . 9 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
46	27, 1, 19, 44, 45	lmodvsdi 17747	. . . . . . . 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 1232	. . . . . . 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 2443	. . . . . 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 17941	. . . . . . . . . 10 |- ( ( W e. LMod /\ T e. L /\ U e. L ) -> ( T .(+) U ) e. L )
50	5, 9, 11, 49	syl3anc 1230	. . . . . . . . 9 |- ( ph -> ( T .(+) U ) e. L )
51	50	adantr 463	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( T .(+) U ) e. L )
52	19, 44, 45, 6	lssvscl 17813	. . . . . . . 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 1231	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) e. ( T .(+) U ) )
54	19, 44, 45, 6	lssvscl 17813	. . . . . . . 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 1231	. . . . . . 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 17813	. . . . . . . 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 1231	. . . . . . 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 16934	. . . . . 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 457	. . . 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 2824	. . 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 3442	. . . . . 6 |- ( ph -> ( T .(+) U ) e. (SubGrp ` W ) )
63		eqid 2402	. . . . . . 7 |- ( W ↾s ( T .(+) U ) ) = ( W ↾s ( T .(+) U ) )
64	63	subgbas 16421	. . . . . 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 3009	. . . 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 2842	. . 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 17818	. . . 4 |- ( ( W e. LMod /\ ( T .(+) U ) e. L ) -> ( W ↾s ( T .(+) U ) ) e. LMod )
70	5, 50, 69	syl2anc 659	. . 3 |- ( ph -> ( W ↾s ( T .(+) U ) ) e. LMod )
71		ovex 6262	. . . . 5 |- ( T .(+) U ) e. _V
72	63, 19	resssca 14883	. . . . 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 2402	. . . 4 |- ( Base ` ( W ↾s ( T .(+) U ) ) ) = ( Base ` ( W ↾s ( T .(+) U ) ) )
75	63, 44	ressvsca 14884	. . . . 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 17886	. . 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 659	. 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 1180	1 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2753   _Vcvv 3058   i^i cin 3412   C_ wss 3413   {csn 3971   ` cfv 5525  (class class class)co 6234   Basecbs 14733   ↾_s cress 14734   +g cplusg 14801  Scalarcsca 14804   .scvsca 14805   0gc0g 14946  SubGrpcsubg 16411   GrpHom cghm 16480  Cntzccntz 16569   LSSumclsm 16870   proj1cpj1 16871   Abelcabl 17015   LModclmod 17724   LSubSpclss 17790   LMHom clmhm 17877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-ndx 14736  df-slot 14737  df-base 14738  df-sets 14739  df-ress 14740  df-plusg 14814  df-sca 14817  df-vsca 14818  df-0g 14948  df-mgm 16088  df-sgrp 16127  df-mnd 16137  df-submnd 16183  df-grp 16273  df-minusg 16274  df-sbg 16275  df-subg 16414  df-ghm 16481  df-cntz 16571  df-lsm 16872  df-pj1 16873  df-cmn 17016  df-abl 17017  df-mgp 17354  df-ur 17366  df-ring 17412  df-lmod 17726  df-lss 17791  df-lmhm 17880

This theorem is referenced by:  pj1lmhm2  17959  pjff  18933