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Theorem pj1id 18838
 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 18297 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
4 eqid 2798 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
54subgss 18293 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
61, 5syl 17 . . . . . 6 (𝜑𝑇 ⊆ (Base‘𝐺))
7 pj1eu.3 . . . . . . 7 (𝜑𝑈 ∈ (SubGrp‘𝐺))
84subgss 18293 . . . . . . 7 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
97, 8syl 17 . . . . . 6 (𝜑𝑈 ⊆ (Base‘𝐺))
103, 6, 93jca 1125 . . . . 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 18834 . . . . 5 (((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) = (𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦)))
1510, 14sylan 583 . . . 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 18835 . . . . 5 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))
21 riotacl2 7119 . . . . 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 2890 . . 3 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)})
24 oveq1 7152 . . . . . . 7 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
2524eqeq2d 2809 . . . . . 6 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2625rexbidv 3257 . . . . 5 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2726elrab 3630 . . . 4 (((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)} ↔ (((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇 ∧ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2827simprbi 500 . . 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 772 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
313ad2antrr 725 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
329ad2antrr 725 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
336ad2antrr 725 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34 simplr 768 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 𝑈))
3512, 17lsmcom2 18793 . . . . . . . . 9 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
361, 7, 19, 35syl3anc 1368 . . . . . . . 8 (𝜑 → (𝑇 𝑈) = (𝑈 𝑇))
3736ad2antrr 725 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 𝑈) = (𝑈 𝑇))
3834, 37eleqtrd 2892 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 𝑇))
394, 11, 12, 13pj1val 18834 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑈 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑈 𝑇)) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4031, 32, 33, 38, 39syl31anc 1370 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4111, 12, 16, 17, 1, 7, 18, 19, 13pj1f 18836 . . . . . . . . 9 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4241ad2antrr 725 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4342, 34ffvelrnd 6839 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
4419ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍𝑈))
4544, 43sseldd 3918 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈))
46 simprl 770 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦𝑈)
4711, 17cntzi 18472 . . . . . . . . 9 ((((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈) ∧ 𝑦𝑈) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
4845, 46, 47syl2anc 587 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + 𝑦) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
4930, 48eqtrd 2833 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50 oveq2 7153 . . . . . . . 8 (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
5150rspceeqv 3587 . . . . . . 7 ((((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋))) → ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣))
5243, 49, 51syl2anc 587 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣))
53 simpll 766 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54 incom 4131 . . . . . . . . . 10 (𝑈𝑇) = (𝑇𝑈)
5554, 18syl5eq 2845 . . . . . . . . 9 (𝜑 → (𝑈𝑇) = { 0 })
5617, 1, 7, 19cntzrecd 18817 . . . . . . . . 9 (𝜑𝑈 ⊆ (𝑍𝑇))
5711, 12, 16, 17, 7, 1, 55, 56pj1eu 18835 . . . . . . . 8 ((𝜑𝑋 ∈ (𝑈 𝑇)) → ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣))
5853, 38, 57syl2anc 587 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣))
59 oveq1 7152 . . . . . . . . . 10 (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
6059eqeq2d 2809 . . . . . . . . 9 (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
6160rexbidv 3257 . . . . . . . 8 (𝑢 = 𝑦 → (∃𝑣𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣)))
6261riota2 7128 . . . . . . 7 ((𝑦𝑈 ∧ ∃!𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) → (∃𝑣𝑇 𝑋 = (𝑦 + 𝑣) ↔ (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
6346, 58, 62syl2anc 587 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (∃𝑣𝑇 𝑋 = (𝑦 + 𝑣) ↔ (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦))
6452, 63mpbid 235 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦)
6540, 64eqtrd 2833 . . . 4 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
6665oveq2d 7161 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
6730, 66eqtr4d 2836 . 2 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
6829, 67rexlimddv 3251 1 ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  ∃!wreu 3108  {crab 3110   ∩ cin 3882   ⊆ wss 3883  {csn 4528  ⟶wf 6328  ‘cfv 6332  ℩crio 7102  (class class class)co 7145  Basecbs 16495  +gcplusg 16577  0gc0g 16725  Grpcgrp 18115  SubGrpcsubg 18286  Cntzccntz 18458  LSSumclsm 18772  proj1cpj1 18773 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-er 8290  df-en 8511  df-dom 8512  df-sdom 8513  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-nn 11644  df-2 11706  df-ndx 16498  df-slot 16499  df-base 16501  df-sets 16502  df-ress 16503  df-plusg 16590  df-0g 16727  df-mgm 17864  df-sgrp 17913  df-mnd 17924  df-grp 18118  df-minusg 18119  df-sbg 18120  df-subg 18289  df-cntz 18460  df-lsm 18774  df-pj1 18775 This theorem is referenced by:  pj1eq  18839  pj1ghm  18842  pj1lmhm  19886
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