Theorem pj1lmhm 17526

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 2467	. . 3 |- ( +g ` W ) = ( +g ` W )
2		pj1lmhm.s	. . 3 |- .(+) = ( LSSum ` W )
3		pj1lmhm.z	. . 3 |- .0. = ( 0g ` W )
4		eqid 2467	. . 3 |- (Cntz ` W ) = (Cntz ` W )
5		pj1lmhm.1	. . . . 5 |- ( ph -> W e. LMod )
6		pj1lmhm.l	. . . . . 6 |- L = ( LSubSp ` W )
7	6	lsssssubg 17384	. . . . 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 3505	. . 3 |- ( ph -> T e. (SubGrp ` W ) )
11		pj1lmhm.3	. . . 4 |- ( ph -> U e. L )
12	8, 11	sseldd 3505	. . 3 |- ( ph -> U e. (SubGrp ` W ) )
13		pj1lmhm.4	. . 3 |- ( ph -> ( T i^i U ) = { .0. } )
14		lmodabl 17337	. . . . 5 |- ( W e. LMod -> W e. Abel )
15	5, 14	syl 16	. . . 4 |- ( ph -> W e. Abel )
16	4, 15, 10, 12	ablcntzd 16653	. . 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 16514	. 2 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) GrpHom W ) )
19		eqid 2467	. . 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 16510	. . . . . . . . 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 6298	. . . . . . 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 2467	. . . . . . . . . . 11 |- ( Base ` W ) = ( Base ` W )
28	27, 6	lssss 17363	. . . . . . . . . 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 16508	. . . . . . . . . 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 6020	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( T P U ) ` y ) e. T )
37	29, 36	sseldd 3505	. . . . . . . 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 17363	. . . . . . . . . 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 16509	. . . . . . . . . 10 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( U P T ) : ( T .(+) U ) --> U )
42	41, 35	ffvelrnd 6020	. . . . . . . . 9 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. U )
43	40, 42	sseldd 3505	. . . . . . . 8 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( ( U P T ) ` y ) e. ( Base ` W ) )
44		eqid 2467	. . . . . . . . 9 |- ( .s ` W ) = ( .s ` W )
45		eqid 2467	. . . . . . . . 9 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
46	27, 1, 19, 44, 45	lmodvsdi 17315	. . . . . . . 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 1230	. . . . . . 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 2508	. . . . . 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 17509	. . . . . . . . . 10 |- ( ( W e. LMod /\ T e. L /\ U e. L ) -> ( T .(+) U ) e. L )
50	5, 9, 11, 49	syl3anc 1228	. . . . . . . . 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 17381	. . . . . . . 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 1229	. . . . . . 7 |- ( ( ph /\ ( x e. ( Base ` (Scalar ` W ) ) /\ y e. ( T .(+) U ) ) ) -> ( x ( .s ` W ) y ) e. ( T .(+) U ) )
54	19, 44, 45, 6	lssvscl 17381	. . . . . . . 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 1229	. . . . . . 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 17381	. . . . . . . 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 1229	. . . . . . 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 16511	. . . . . 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 2885	. . 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 3505	. . . . . 6 |- ( ph -> ( T .(+) U ) e. (SubGrp ` W ) )
63		eqid 2467	. . . . . . 7 |- ( W ↾s ( T .(+) U ) ) = ( W ↾s ( T .(+) U ) )
64	63	subgbas 15997	. . . . . 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 3064	. . . 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 2903	. . 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 17386	. . . 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 6307	. . . . 5 |- ( T .(+) U ) e. _V
72	63, 19	resssca 14626	. . . . 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 2467	. . . 4 |- ( Base ` ( W ↾s ( T .(+) U ) ) ) = ( Base ` ( W ↾s ( T .(+) U ) ) )
75	63, 44	ressvsca 14627	. . . . 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 17454	. . 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 1179	1 |- ( ph -> ( T P U ) e. ( ( W ↾s ( T .(+) U ) ) LMHom W ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   i^i cin 3475   C_ wss 3476   {csn 4027   ` cfv 5586  (class class class)co 6282   Basecbs 14483   ↾_s cress 14484   +g cplusg 14548  Scalarcsca 14551   .scvsca 14552   0gc0g 14688  SubGrpcsubg 15987   GrpHom cghm 16056  Cntzccntz 16145   LSSumclsm 16447   proj1cpj1 16448   Abelcabl 16592   LModclmod 17292   LSubSpclss 17358   LMHom clmhm 17445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-sca 14564  df-vsca 14565  df-0g 14690  df-mnd 15725  df-submnd 15775  df-grp 15855  df-minusg 15856  df-sbg 15857  df-subg 15990  df-ghm 16057  df-cntz 16147  df-lsm 16449  df-pj1 16450  df-cmn 16593  df-abl 16594  df-mgp 16929  df-ur 16941  df-rng 16985  df-lmod 17294  df-lss 17359  df-lmhm 17448

This theorem is referenced by:  pj1lmhm2  17527  pjff  18507