Theorem pj1id 16506

Description: Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
pj1eu.a	|- .+ = ( +g ` G )
pj1eu.s	|- .(+) = ( LSSum ` G )
pj1eu.o	|- .0. = ( 0g ` G )
pj1eu.z	|- Z = (Cntz ` G )
pj1eu.2	|- ( ph -> T e. (SubGrp ` G ) )
pj1eu.3	|- ( ph -> U e. (SubGrp ` G ) )
pj1eu.4	|- ( ph -> ( T i^i U ) = { .0. } )
pj1eu.5	|- ( ph -> T C_ ( Z ` U ) )
pj1f.p	|- P = ( proj1 ` G )

Assertion

Ref	Expression
pj1id	|- ( ( ph /\ X e. ( T .(+) U ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) )

Proof of Theorem pj1id

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

Step	Hyp	Ref	Expression
1		pj1eu.2	. . . . . . 7 |- ( ph -> T e. (SubGrp ` G ) )
2		subgrcl 15994	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 16	. . . . . 6 |- ( ph -> G e. Grp )
4		eqid 2460	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 15990	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> T C_ ( Base ` G ) )
6	1, 5	syl 16	. . . . . 6 |- ( ph -> T C_ ( Base ` G ) )
7		pj1eu.3	. . . . . . 7 |- ( ph -> U e. (SubGrp ` G ) )
8	4	subgss 15990	. . . . . . 7 |- ( U e. (SubGrp ` G ) -> U C_ ( Base ` G ) )
9	7, 8	syl 16	. . . . . 6 |- ( ph -> U C_ ( Base ` G ) )
10	3, 6, 9	3jca 1171	. . . . 5 |- ( ph -> ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) )
11		pj1eu.a	. . . . . 6 |- .+ = ( +g ` G )
12		pj1eu.s	. . . . . 6 |- .(+) = ( LSSum ` G )
13		pj1f.p	. . . . . 6 |- P = ( proj1 ` G )
14	4, 11, 12, 13	pj1val 16502	. . . . 5 |- ( ( ( G e. Grp /\ T C_ ( Base ` G ) /\ U C_ ( Base ` G ) ) /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) = ( iota_ x e. T E. y e. U X = ( x .+ y ) ) )
15	10, 14	sylan 471	. . . 4 |- ( ( ph /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) = ( iota_ x e. T E. y e. U X = ( x .+ y ) ) )
16		pj1eu.o	. . . . . 6 |- .0. = ( 0g ` G )
17		pj1eu.z	. . . . . 6 |- Z = (Cntz ` G )
18		pj1eu.4	. . . . . 6 |- ( ph -> ( T i^i U ) = { .0. } )
19		pj1eu.5	. . . . . 6 |- ( ph -> T C_ ( Z ` U ) )
20	11, 12, 16, 17, 1, 7, 18, 19	pj1eu 16503	. . . . 5 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E! x e. T E. y e. U X = ( x .+ y ) )
21		riotacl2 6250	. . . . 5 |- ( E! x e. T E. y e. U X = ( x .+ y ) -> ( iota_ x e. T E. y e. U X = ( x .+ y ) ) e. { x e. T | E. y e. U X = ( x .+ y ) } )
22	20, 21	syl 16	. . . 4 |- ( ( ph /\ X e. ( T .(+) U ) ) -> ( iota_ x e. T E. y e. U X = ( x .+ y ) ) e. { x e. T | E. y e. U X = ( x .+ y ) } )
23	15, 22	eqeltrd 2548	. . 3 |- ( ( ph /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) e. { x e. T | E. y e. U X = ( x .+ y ) } )
24		oveq1 6282	. . . . . . 7 |- ( x = ( ( T P U ) ` X ) -> ( x .+ y ) = ( ( ( T P U ) ` X ) .+ y ) )
25	24	eqeq2d 2474	. . . . . 6 |- ( x = ( ( T P U ) ` X ) -> ( X = ( x .+ y ) <-> X = ( ( ( T P U ) ` X ) .+ y ) ) )
26	25	rexbidv 2966	. . . . 5 |- ( x = ( ( T P U ) ` X ) -> ( E. y e. U X = ( x .+ y ) <-> E. y e. U X = ( ( ( T P U ) ` X ) .+ y ) ) )
27	26	elrab 3254	. . . 4 |- ( ( ( T P U ) ` X ) e. { x e. T | E. y e. U X = ( x .+ y ) } <-> ( ( ( T P U ) ` X ) e. T /\ E. y e. U X = ( ( ( T P U ) ` X ) .+ y ) ) )
28	27	simprbi 464	. . 3 |- ( ( ( T P U ) ` X ) e. { x e. T | E. y e. U X = ( x .+ y ) } -> E. y e. U X = ( ( ( T P U ) ` X ) .+ y ) )
29	23, 28	syl 16	. 2 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E. y e. U X = ( ( ( T P U ) ` X ) .+ y ) )
30		simprr 756	. . 3 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( ( ( T P U ) ` X ) .+ y ) )
31	3	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> G e. Grp )
32	9	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> U C_ ( Base ` G ) )
33	6	ad2antrr 725	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Base ` G ) )
34		simplr 754	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X e. ( T .(+) U ) )
35	12, 17	lsmcom2 16464	. . . . . . . . 9 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
36	1, 7, 19, 35	syl3anc 1223	. . . . . . . 8 |- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
37	36	ad2antrr 725	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( T .(+) U ) = ( U .(+) T ) )
38	34, 37	eleqtrd 2550	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X e. ( U .(+) T ) )
39	4, 11, 12, 13	pj1val 16502	. . . . . 6 |- ( ( ( G e. Grp /\ U C_ ( Base ` G ) /\ T C_ ( Base ` G ) ) /\ X e. ( U .(+) T ) ) -> ( ( U P T ) ` X ) = ( iota_ u e. U E. v e. T X = ( u .+ v ) ) )
40	31, 32, 33, 38, 39	syl31anc 1226	. . . . 5 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( U P T ) ` X ) = ( iota_ u e. U E. v e. T X = ( u .+ v ) ) )
41	11, 12, 16, 17, 1, 7, 18, 19, 13	pj1f 16504	. . . . . . . . 9 |- ( ph -> ( T P U ) : ( T .(+) U ) --> T )
42	41	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( T P U ) : ( T .(+) U ) --> T )
43	42, 34	ffvelrnd 6013	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. T )
44	19	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Z ` U ) )
45	44, 43	sseldd 3498	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. ( Z ` U ) )
46		simprl 755	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> y e. U )
47	11, 17	cntzi 16155	. . . . . . . . 9 |- ( ( ( ( T P U ) ` X ) e. ( Z ` U ) /\ y e. U ) -> ( ( ( T P U ) ` X ) .+ y ) = ( y .+ ( ( T P U ) ` X ) ) )
48	45, 46, 47	syl2anc 661	. . . . . . . 8 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( ( T P U ) ` X ) .+ y ) = ( y .+ ( ( T P U ) ` X ) ) )
49	30, 48	eqtrd 2501	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( y .+ ( ( T P U ) ` X ) ) )
50		oveq2 6283	. . . . . . . . 9 |- ( v = ( ( T P U ) ` X ) -> ( y .+ v ) = ( y .+ ( ( T P U ) ` X ) ) )
51	50	eqeq2d 2474	. . . . . . . 8 |- ( v = ( ( T P U ) ` X ) -> ( X = ( y .+ v ) <-> X = ( y .+ ( ( T P U ) ` X ) ) ) )
52	51	rspcev 3207	. . . . . . 7 |- ( ( ( ( T P U ) ` X ) e. T /\ X = ( y .+ ( ( T P U ) ` X ) ) ) -> E. v e. T X = ( y .+ v ) )
53	43, 49, 52	syl2anc 661	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> E. v e. T X = ( y .+ v ) )
54		simpll 753	. . . . . . . 8 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ph )
55		incom 3684	. . . . . . . . . 10 |- ( U i^i T ) = ( T i^i U )
56	55, 18	syl5eq 2513	. . . . . . . . 9 |- ( ph -> ( U i^i T ) = { .0. } )
57	17, 1, 7, 19	cntzrecd 16485	. . . . . . . . 9 |- ( ph -> U C_ ( Z ` T ) )
58	11, 12, 16, 17, 7, 1, 56, 57	pj1eu 16503	. . . . . . . 8 |- ( ( ph /\ X e. ( U .(+) T ) ) -> E! u e. U E. v e. T X = ( u .+ v ) )
59	54, 38, 58	syl2anc 661	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> E! u e. U E. v e. T X = ( u .+ v ) )
60		oveq1 6282	. . . . . . . . . 10 |- ( u = y -> ( u .+ v ) = ( y .+ v ) )
61	60	eqeq2d 2474	. . . . . . . . 9 |- ( u = y -> ( X = ( u .+ v ) <-> X = ( y .+ v ) ) )
62	61	rexbidv 2966	. . . . . . . 8 |- ( u = y -> ( E. v e. T X = ( u .+ v ) <-> E. v e. T X = ( y .+ v ) ) )
63	62	riota2 6259	. . . . . . 7 |- ( ( y e. U /\ E! u e. U E. v e. T X = ( u .+ v ) ) -> ( E. v e. T X = ( y .+ v ) <-> ( iota_ u e. U E. v e. T X = ( u .+ v ) ) = y ) )
64	46, 59, 63	syl2anc 661	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( E. v e. T X = ( y .+ v ) <-> ( iota_ u e. U E. v e. T X = ( u .+ v ) ) = y ) )
65	53, 64	mpbid 210	. . . . 5 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( iota_ u e. U E. v e. T X = ( u .+ v ) ) = y )
66	40, 65	eqtrd 2501	. . . 4 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( U P T ) ` X ) = y )
67	66	oveq2d 6291	. . 3 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) = ( ( ( T P U ) ` X ) .+ y ) )
68	30, 67	eqtr4d 2504	. 2 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) )
69	29, 68	rexlimddv 2952	1 |- ( ( ph /\ X e. ( T .(+) U ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   E.wrex 2808   E!wreu 2809   {crab 2811   i^i cin 3468   C_ wss 3469   {csn 4020   -->wf 5575   ` cfv 5579   iota_crio 6235  (class class class)co 6275   Basecbs 14479   +g cplusg 14544   0gc0g 14684   Grpcgrp 15716  SubGrpcsubg 15983  Cntzccntz 16141   LSSumclsm 16443   proj1cpj1 16444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-0g 14686  df-mnd 15721  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-pj1 16446

This theorem is referenced by:  pj1eq  16507  pj1ghm  16510  pj1lmhm  17522