Theorem pj1id 17284

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 16773	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> G e. Grp )
4		eqid 2429	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 16769	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> T C_ ( Base ` G ) )
6	1, 5	syl 17	. . . . . 6 |- ( ph -> T C_ ( Base ` G ) )
7		pj1eu.3	. . . . . . 7 |- ( ph -> U e. (SubGrp ` G ) )
8	4	subgss 16769	. . . . . . 7 |- ( U e. (SubGrp ` G ) -> U C_ ( Base ` G ) )
9	7, 8	syl 17	. . . . . 6 |- ( ph -> U C_ ( Base ` G ) )
10	3, 6, 9	3jca 1185	. . . . 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 17280	. . . . 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 473	. . . 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 17281	. . . . 5 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E! x e. T E. y e. U X = ( x .+ y ) )
21		riotacl2 6280	. . . . 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 17	. . . 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 2517	. . 3 |- ( ( ph /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) e. { x e. T | E. y e. U X = ( x .+ y ) } )
24		oveq1 6312	. . . . . . 7 |- ( x = ( ( T P U ) ` X ) -> ( x .+ y ) = ( ( ( T P U ) ` X ) .+ y ) )
25	24	eqeq2d 2443	. . . . . 6 |- ( x = ( ( T P U ) ` X ) -> ( X = ( x .+ y ) <-> X = ( ( ( T P U ) ` X ) .+ y ) ) )
26	25	rexbidv 2946	. . . . 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 3235	. . . 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 465	. . 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 17	. 2 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E. y e. U X = ( ( ( T P U ) ` X ) .+ y ) )
30		simprr 764	. . 3 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( ( ( T P U ) ` X ) .+ y ) )
31	3	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> G e. Grp )
32	9	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> U C_ ( Base ` G ) )
33	6	ad2antrr 730	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Base ` G ) )
34		simplr 760	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X e. ( T .(+) U ) )
35	12, 17	lsmcom2 17242	. . . . . . . . 9 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
36	1, 7, 19, 35	syl3anc 1264	. . . . . . . 8 |- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
37	36	ad2antrr 730	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( T .(+) U ) = ( U .(+) T ) )
38	34, 37	eleqtrd 2519	. . . . . 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 17280	. . . . . 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 1267	. . . . 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 17282	. . . . . . . . 9 |- ( ph -> ( T P U ) : ( T .(+) U ) --> T )
42	41	ad2antrr 730	. . . . . . . 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 6038	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. T )
44	19	ad2antrr 730	. . . . . . . . . 10 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Z ` U ) )
45	44, 43	sseldd 3471	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. ( Z ` U ) )
46		simprl 762	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> y e. U )
47	11, 17	cntzi 16934	. . . . . . . . 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 665	. . . . . . . 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 2470	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( y .+ ( ( T P U ) ` X ) ) )
50		oveq2 6313	. . . . . . . . 9 |- ( v = ( ( T P U ) ` X ) -> ( y .+ v ) = ( y .+ ( ( T P U ) ` X ) ) )
51	50	eqeq2d 2443	. . . . . . . 8 |- ( v = ( ( T P U ) ` X ) -> ( X = ( y .+ v ) <-> X = ( y .+ ( ( T P U ) ` X ) ) ) )
52	51	rspcev 3188	. . . . . . 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 665	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> E. v e. T X = ( y .+ v ) )
54		simpll 758	. . . . . . . 8 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ph )
55		incom 3661	. . . . . . . . . 10 |- ( U i^i T ) = ( T i^i U )
56	55, 18	syl5eq 2482	. . . . . . . . 9 |- ( ph -> ( U i^i T ) = { .0. } )
57	17, 1, 7, 19	cntzrecd 17263	. . . . . . . . 9 |- ( ph -> U C_ ( Z ` T ) )
58	11, 12, 16, 17, 7, 1, 56, 57	pj1eu 17281	. . . . . . . 8 |- ( ( ph /\ X e. ( U .(+) T ) ) -> E! u e. U E. v e. T X = ( u .+ v ) )
59	54, 38, 58	syl2anc 665	. . . . . . 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 6312	. . . . . . . . . 10 |- ( u = y -> ( u .+ v ) = ( y .+ v ) )
61	60	eqeq2d 2443	. . . . . . . . 9 |- ( u = y -> ( X = ( u .+ v ) <-> X = ( y .+ v ) ) )
62	61	rexbidv 2946	. . . . . . . 8 |- ( u = y -> ( E. v e. T X = ( u .+ v ) <-> E. v e. T X = ( y .+ v ) ) )
63	62	riota2 6289	. . . . . . 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 665	. . . . . 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 213	. . . . 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 2470	. . . 4 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( U P T ) ` X ) = y )
67	66	oveq2d 6321	. . 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 2473	. 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 2928	1 |- ( ( ph /\ X e. ( T .(+) U ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1870   E.wrex 2783   E!wreu 2784   {crab 2786   i^i cin 3441   C_ wss 3442   {csn 4002   -->wf 5597   ` cfv 5601   iota_crio 6266  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   0gc0g 15297   Grpcgrp 16620  SubGrpcsubg 16762  Cntzccntz 16920   LSSumclsm 17221   proj1cpj1 17222

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-lsm 17223  df-pj1 17224

This theorem is referenced by:  pj1eq  17285  pj1ghm  17288  pj1lmhm  18258