Theorem pj1id 18112

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 17599	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> G e. Grp )
4		eqid 2622	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 17595	. . . . . . 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 17595	. . . . . . 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 1242	. . . . 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 18108	. . . . 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 488	. . . 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 18109	. . . . 5 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E! x e. T E. y e. U X = ( x .+ y ) )
21		riotacl2 6624	. . . . 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 2701	. . 3 |- ( ( ph /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) e. { x e. T | E. y e. U X = ( x .+ y ) } )
24		oveq1 6657	. . . . . . 7 |- ( x = ( ( T P U ) ` X ) -> ( x .+ y ) = ( ( ( T P U ) ` X ) .+ y ) )
25	24	eqeq2d 2632	. . . . . 6 |- ( x = ( ( T P U ) ` X ) -> ( X = ( x .+ y ) <-> X = ( ( ( T P U ) ` X ) .+ y ) ) )
26	25	rexbidv 3052	. . . . 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 3363	. . . 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 480	. . 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 796	. . 3 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( ( ( T P U ) ` X ) .+ y ) )
31	3	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> G e. Grp )
32	9	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> U C_ ( Base ` G ) )
33	6	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Base ` G ) )
34		simplr 792	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X e. ( T .(+) U ) )
35	12, 17	lsmcom2 18070	. . . . . . . . 9 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
36	1, 7, 19, 35	syl3anc 1326	. . . . . . . 8 |- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
37	36	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( T .(+) U ) = ( U .(+) T ) )
38	34, 37	eleqtrd 2703	. . . . . 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 18108	. . . . . 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 1329	. . . . 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 18110	. . . . . . . . 9 |- ( ph -> ( T P U ) : ( T .(+) U ) --> T )
42	41	ad2antrr 762	. . . . . . . 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 6360	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. T )
44	19	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Z ` U ) )
45	44, 43	sseldd 3604	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. ( Z ` U ) )
46		simprl 794	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> y e. U )
47	11, 17	cntzi 17762	. . . . . . . . 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 693	. . . . . . . 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 2656	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( y .+ ( ( T P U ) ` X ) ) )
50		oveq2 6658	. . . . . . . . 9 |- ( v = ( ( T P U ) ` X ) -> ( y .+ v ) = ( y .+ ( ( T P U ) ` X ) ) )
51	50	eqeq2d 2632	. . . . . . . 8 |- ( v = ( ( T P U ) ` X ) -> ( X = ( y .+ v ) <-> X = ( y .+ ( ( T P U ) ` X ) ) ) )
52	51	rspcev 3309	. . . . . . 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 693	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> E. v e. T X = ( y .+ v ) )
54		simpll 790	. . . . . . . 8 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ph )
55		incom 3805	. . . . . . . . . 10 |- ( U i^i T ) = ( T i^i U )
56	55, 18	syl5eq 2668	. . . . . . . . 9 |- ( ph -> ( U i^i T ) = { .0. } )
57	17, 1, 7, 19	cntzrecd 18091	. . . . . . . . 9 |- ( ph -> U C_ ( Z ` T ) )
58	11, 12, 16, 17, 7, 1, 56, 57	pj1eu 18109	. . . . . . . 8 |- ( ( ph /\ X e. ( U .(+) T ) ) -> E! u e. U E. v e. T X = ( u .+ v ) )
59	54, 38, 58	syl2anc 693	. . . . . . 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 6657	. . . . . . . . . 10 |- ( u = y -> ( u .+ v ) = ( y .+ v ) )
61	60	eqeq2d 2632	. . . . . . . . 9 |- ( u = y -> ( X = ( u .+ v ) <-> X = ( y .+ v ) ) )
62	61	rexbidv 3052	. . . . . . . 8 |- ( u = y -> ( E. v e. T X = ( u .+ v ) <-> E. v e. T X = ( y .+ v ) ) )
63	62	riota2 6633	. . . . . . 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 693	. . . . . 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 222	. . . . 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 2656	. . . 4 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( U P T ) ` X ) = y )
67	66	oveq2d 6666	. . 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 2659	. 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 3035	1 |- ( ( ph /\ X e. ( T .(+) U ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   E!wreu 2914   {crab 2916   i^i cin 3573   C_ wss 3574   {csn 4177   -->wf 5884   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049   proj1cpj1 18050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-pj1 18052

This theorem is referenced by:  pj1eq  18113  pj1ghm  18116  pj1lmhm  19100