Theorem pj1id 17342

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 16815	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> G e. Grp )
3	1, 2	syl 17	. . . . . 6 |- ( ph -> G e. Grp )
4		eqid 2450	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
5	4	subgss 16811	. . . . . . 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 16811	. . . . . . 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 1187	. . . . 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 17338	. . . . 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 474	. . . 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 17339	. . . . 5 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E! x e. T E. y e. U X = ( x .+ y ) )
21		riotacl2 6263	. . . . 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 2528	. . 3 |- ( ( ph /\ X e. ( T .(+) U ) ) -> ( ( T P U ) ` X ) e. { x e. T | E. y e. U X = ( x .+ y ) } )
24		oveq1 6295	. . . . . . 7 |- ( x = ( ( T P U ) ` X ) -> ( x .+ y ) = ( ( ( T P U ) ` X ) .+ y ) )
25	24	eqeq2d 2460	. . . . . 6 |- ( x = ( ( T P U ) ` X ) -> ( X = ( x .+ y ) <-> X = ( ( ( T P U ) ` X ) .+ y ) ) )
26	25	rexbidv 2900	. . . . 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 3195	. . . 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 466	. . 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 765	. . 3 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( ( ( T P U ) ` X ) .+ y ) )
31	3	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> G e. Grp )
32	9	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> U C_ ( Base ` G ) )
33	6	ad2antrr 731	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Base ` G ) )
34		simplr 761	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X e. ( T .(+) U ) )
35	12, 17	lsmcom2 17300	. . . . . . . . 9 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
36	1, 7, 19, 35	syl3anc 1267	. . . . . . . 8 |- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
37	36	ad2antrr 731	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( T .(+) U ) = ( U .(+) T ) )
38	34, 37	eleqtrd 2530	. . . . . 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 17338	. . . . . 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 1270	. . . . 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 17340	. . . . . . . . 9 |- ( ph -> ( T P U ) : ( T .(+) U ) --> T )
42	41	ad2antrr 731	. . . . . . . 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 6021	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. T )
44	19	ad2antrr 731	. . . . . . . . . 10 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> T C_ ( Z ` U ) )
45	44, 43	sseldd 3432	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( T P U ) ` X ) e. ( Z ` U ) )
46		simprl 763	. . . . . . . . 9 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> y e. U )
47	11, 17	cntzi 16976	. . . . . . . . 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 666	. . . . . . . 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 2484	. . . . . . 7 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> X = ( y .+ ( ( T P U ) ` X ) ) )
50		oveq2 6296	. . . . . . . . 9 |- ( v = ( ( T P U ) ` X ) -> ( y .+ v ) = ( y .+ ( ( T P U ) ` X ) ) )
51	50	eqeq2d 2460	. . . . . . . 8 |- ( v = ( ( T P U ) ` X ) -> ( X = ( y .+ v ) <-> X = ( y .+ ( ( T P U ) ` X ) ) ) )
52	51	rspcev 3149	. . . . . . 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 666	. . . . . 6 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> E. v e. T X = ( y .+ v ) )
54		simpll 759	. . . . . . . 8 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ph )
55		incom 3624	. . . . . . . . . 10 |- ( U i^i T ) = ( T i^i U )
56	55, 18	syl5eq 2496	. . . . . . . . 9 |- ( ph -> ( U i^i T ) = { .0. } )
57	17, 1, 7, 19	cntzrecd 17321	. . . . . . . . 9 |- ( ph -> U C_ ( Z ` T ) )
58	11, 12, 16, 17, 7, 1, 56, 57	pj1eu 17339	. . . . . . . 8 |- ( ( ph /\ X e. ( U .(+) T ) ) -> E! u e. U E. v e. T X = ( u .+ v ) )
59	54, 38, 58	syl2anc 666	. . . . . . 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 6295	. . . . . . . . . 10 |- ( u = y -> ( u .+ v ) = ( y .+ v ) )
61	60	eqeq2d 2460	. . . . . . . . 9 |- ( u = y -> ( X = ( u .+ v ) <-> X = ( y .+ v ) ) )
62	61	rexbidv 2900	. . . . . . . 8 |- ( u = y -> ( E. v e. T X = ( u .+ v ) <-> E. v e. T X = ( y .+ v ) ) )
63	62	riota2 6272	. . . . . . 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 666	. . . . . 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 214	. . . . 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 2484	. . . 4 |- ( ( ( ph /\ X e. ( T .(+) U ) ) /\ ( y e. U /\ X = ( ( ( T P U ) ` X ) .+ y ) ) ) -> ( ( U P T ) ` X ) = y )
67	66	oveq2d 6304	. . 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 2487	. 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 2882	1 |- ( ( ph /\ X e. ( T .(+) U ) ) -> X = ( ( ( T P U ) ` X ) .+ ( ( U P T ) ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   E.wrex 2737   E!wreu 2738   {crab 2740   i^i cin 3402   C_ wss 3403   {csn 3967   -->wf 5577   ` cfv 5581   iota_crio 6249  (class class class)co 6288   Basecbs 15114   +g cplusg 15183   0gc0g 15331   Grpcgrp 16662  SubGrpcsubg 16804  Cntzccntz 16962   LSSumclsm 17279   proj1cpj1 17280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-ndx 15117  df-slot 15118  df-base 15119  df-sets 15120  df-ress 15121  df-plusg 15196  df-0g 15333  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-grp 16666  df-minusg 16667  df-sbg 16668  df-subg 16807  df-cntz 16964  df-lsm 17281  df-pj1 17282

This theorem is referenced by:  pj1eq  17343  pj1ghm  17346  pj1lmhm  18316