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Theorem pj1id 18304
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 17792 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
4 eqid 2752 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
54subgss 17788 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
61, 5syl 17 . . . . . 6 (𝜑𝑇 ⊆ (Base‘𝐺))
7 pj1eu.3 . . . . . . 7 (𝜑𝑈 ∈ (SubGrp‘𝐺))
84subgss 17788 . . . . . . 7 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
97, 8syl 17 . . . . . 6 (𝜑𝑈 ⊆ (Base‘𝐺))
103, 6, 93jca 1122 . . . . 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 18300 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
1510, 14sylan 489 . . . 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 18301 . . . . 5 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))
21 riotacl2 6779 . . . . 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 2831 . . 3 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)})
24 oveq1 6812 . . . . . . 7 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
2524eqeq2d 2762 . . . . . 6 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2625rexbidv 3182 . . . . 5 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2726elrab 3496 . . . 4 (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2827simprbi 483 . . 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 813 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
313ad2antrr 764 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
329ad2antrr 764 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
336ad2antrr 764 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34 simplr 809 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 𝑈))
3512, 17lsmcom2 18262 . . . . . . . . 9 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
361, 7, 19, 35syl3anc 1473 . . . . . . . 8 (𝜑 → (𝑇 𝑈) = (𝑈 𝑇))
3736ad2antrr 764 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 𝑈) = (𝑈 𝑇))
3834, 37eleqtrd 2833 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 𝑇))
394, 11, 12, 13pj1val 18300 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑈 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑈 𝑇)) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4031, 32, 33, 38, 39syl31anc 1476 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4111, 12, 16, 17, 1, 7, 18, 19, 13pj1f 18302 . . . . . . . . 9 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4241ad2antrr 764 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4342, 34ffvelrnd 6515 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
4419ad2antrr 764 . . . . . . . . . 10 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍𝑈))
4544, 43sseldd 3737 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈))
46 simprl 811 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦𝑈)
4711, 17cntzi 17954 . . . . . . . . 9 ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈) ∧ 𝑦𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
4845, 46, 47syl2anc 696 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
4930, 48eqtrd 2786 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50 oveq2 6813 . . . . . . . . 9 (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
5150eqeq2d 2762 . . . . . . . 8 (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑦 + 𝑣) ↔ 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))))
5251rspcev 3441 . . . . . . 7 ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣))
5343, 49, 52syl2anc 696 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣))
54 simpll 807 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
55 incom 3940 . . . . . . . . . 10 (𝑈𝑇) = (𝑇𝑈)
5655, 18syl5eq 2798 . . . . . . . . 9 (𝜑 → (𝑈𝑇) = { 0 })
5717, 1, 7, 19cntzrecd 18283 . . . . . . . . 9 (𝜑𝑈 ⊆ (𝑍𝑇))
5811, 12, 16, 17, 7, 1, 56, 57pj1eu 18301 . . . . . . . 8 ((𝜑𝑋 ∈ (𝑈 𝑇)) → ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣))
5954, 38, 58syl2anc 696 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣))
60 oveq1 6812 . . . . . . . . . 10 (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
6160eqeq2d 2762 . . . . . . . . 9 (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
6261rexbidv 3182 . . . . . . . 8 (𝑢 = 𝑦 → (∃𝑣𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣)))
6362riota2 6788 . . . . . . 7 ((𝑦𝑈 ∧ ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) → (∃𝑣𝑇 𝑋 = (𝑦 + 𝑣) ↔ (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
6446, 59, 63syl2anc 696 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (∃𝑣𝑇 𝑋 = (𝑦 + 𝑣) ↔ (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
6553, 64mpbid 222 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦)
6640, 65eqtrd 2786 . . . 4 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
6766oveq2d 6821 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
6830, 67eqtr4d 2789 . 2 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
6929, 68rexlimddv 3165 1 ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wrex 3043  ∃!wreu 3044  {crab 3046  cin 3706  wss 3707  {csn 4313  wf 6037  cfv 6041  crio 6765  (class class class)co 6805  Basecbs 16051  +gcplusg 16135  0gc0g 16294  Grpcgrp 17615  SubGrpcsubg 17781  Cntzccntz 17940  LSSumclsm 18241  proj1cpj1 18242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-er 7903  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-nn 11205  df-2 11263  df-ndx 16054  df-slot 16055  df-base 16057  df-sets 16058  df-ress 16059  df-plusg 16148  df-0g 16296  df-mgm 17435  df-sgrp 17477  df-mnd 17488  df-grp 17618  df-minusg 17619  df-sbg 17620  df-subg 17784  df-cntz 17942  df-lsm 18243  df-pj1 18244
This theorem is referenced by:  pj1eq  18305  pj1ghm  18308  pj1lmhm  19294
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