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Theorem pj1id 19481
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 + = (+g𝐺)
pj1eu.s = (LSSum‘𝐺)
pj1eu.o 0 = (0g𝐺)
pj1eu.z 𝑍 = (Cntz‘𝐺)
pj1eu.2 (𝜑𝑇 ∈ (SubGrp‘𝐺))
pj1eu.3 (𝜑𝑈 ∈ (SubGrp‘𝐺))
pj1eu.4 (𝜑 → (𝑇𝑈) = { 0 })
pj1eu.5 (𝜑𝑇 ⊆ (𝑍𝑈))
pj1f.p 𝑃 = (proj1𝐺)
Assertion
Ref Expression
pj1id ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))

Proof of Theorem pj1id
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pj1eu.2 . . . . . . 7 (𝜑𝑇 ∈ (SubGrp‘𝐺))
2 subgrcl 18933 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
4 eqid 2736 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
54subgss 18929 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
61, 5syl 17 . . . . . 6 (𝜑𝑇 ⊆ (Base‘𝐺))
7 pj1eu.3 . . . . . . 7 (𝜑𝑈 ∈ (SubGrp‘𝐺))
84subgss 18929 . . . . . . 7 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
97, 8syl 17 . . . . . 6 (𝜑𝑈 ⊆ (Base‘𝐺))
103, 6, 93jca 1128 . . . . 5 (𝜑 → (𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)))
11 pj1eu.a . . . . . 6 + = (+g𝐺)
12 pj1eu.s . . . . . 6 = (LSSum‘𝐺)
13 pj1f.p . . . . . 6 𝑃 = (proj1𝐺)
144, 11, 12, 13pj1val 19477 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
1510, 14sylan 580 . . . 4 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
16 pj1eu.o . . . . . 6 0 = (0g𝐺)
17 pj1eu.z . . . . . 6 𝑍 = (Cntz‘𝐺)
18 pj1eu.4 . . . . . 6 (𝜑 → (𝑇𝑈) = { 0 })
19 pj1eu.5 . . . . . 6 (𝜑𝑇 ⊆ (𝑍𝑈))
2011, 12, 16, 17, 1, 7, 18, 19pj1eu 19478 . . . . 5 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))
21 riotacl2 7330 . . . . 5 (∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦) → (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)})
2220, 21syl 17 . . . 4 ((𝜑𝑋 ∈ (𝑇 𝑈)) → (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)})
2315, 22eqeltrd 2838 . . 3 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)})
24 oveq1 7364 . . . . . . 7 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
2524eqeq2d 2747 . . . . . 6 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2625rexbidv 3175 . . . . 5 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2726elrab 3645 . . . 4 (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2827simprbi 497 . . 3 (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)} → ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
2923, 28syl 17 . 2 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
30 simprr 771 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
313ad2antrr 724 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
329ad2antrr 724 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
336ad2antrr 724 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34 simplr 767 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 𝑈))
3512, 17lsmcom2 19437 . . . . . . . . 9 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
361, 7, 19, 35syl3anc 1371 . . . . . . . 8 (𝜑 → (𝑇 𝑈) = (𝑈 𝑇))
3736ad2antrr 724 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 𝑈) = (𝑈 𝑇))
3834, 37eleqtrd 2840 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 𝑇))
394, 11, 12, 13pj1val 19477 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑈 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑈 𝑇)) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4031, 32, 33, 38, 39syl31anc 1373 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4111, 12, 16, 17, 1, 7, 18, 19, 13pj1f 19479 . . . . . . . . 9 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4241ad2antrr 724 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4342, 34ffvelcdmd 7036 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
4419ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍𝑈))
4544, 43sseldd 3945 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈))
46 simprl 769 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦𝑈)
4711, 17cntzi 19109 . . . . . . . . 9 ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈) ∧ 𝑦𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
4845, 46, 47syl2anc 584 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
4930, 48eqtrd 2776 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50 oveq2 7365 . . . . . . . 8 (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
5150rspceeqv 3595 . . . . . . 7 ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣))
5243, 49, 51syl2anc 584 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣))
53 simpll 765 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54 incom 4161 . . . . . . . . . 10 (𝑈𝑇) = (𝑇𝑈)
5554, 18eqtrid 2788 . . . . . . . . 9 (𝜑 → (𝑈𝑇) = { 0 })
5617, 1, 7, 19cntzrecd 19460 . . . . . . . . 9 (𝜑𝑈 ⊆ (𝑍𝑇))
5711, 12, 16, 17, 7, 1, 55, 56pj1eu 19478 . . . . . . . 8 ((𝜑𝑋 ∈ (𝑈 𝑇)) → ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣))
5853, 38, 57syl2anc 584 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣))
59 oveq1 7364 . . . . . . . . . 10 (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
6059eqeq2d 2747 . . . . . . . . 9 (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
6160rexbidv 3175 . . . . . . . 8 (𝑢 = 𝑦 → (∃𝑣𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣)))
6261riota2 7339 . . . . . . 7 ((𝑦𝑈 ∧ ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) → (∃𝑣𝑇 𝑋 = (𝑦 + 𝑣) ↔ (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
6346, 58, 62syl2anc 584 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (∃𝑣𝑇 𝑋 = (𝑦 + 𝑣) ↔ (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
6452, 63mpbid 231 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦)
6540, 64eqtrd 2776 . . . 4 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
6665oveq2d 7373 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
6730, 66eqtr4d 2779 . 2 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
6829, 67rexlimddv 3158 1 ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  ∃!wreu 3351  {crab 3407  cin 3909  wss 3910  {csn 4586  wf 6492  cfv 6496  crio 7312  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  SubGrpcsubg 18922  Cntzccntz 19095  LSSumclsm 19416  proj1cpj1 19417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-pj1 19419
This theorem is referenced by:  pj1eq  19482  pj1ghm  19485  pj1lmhm  20561
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