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Theorem pj1id 19372
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 18827 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
4 eqid 2737 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
54subgss 18823 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
61, 5syl 17 . . . . . 6 (𝜑𝑇 ⊆ (Base‘𝐺))
7 pj1eu.3 . . . . . . 7 (𝜑𝑈 ∈ (SubGrp‘𝐺))
84subgss 18823 . . . . . . 7 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
97, 8syl 17 . . . . . 6 (𝜑𝑈 ⊆ (Base‘𝐺))
103, 6, 93jca 1127 . . . . 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 19368 . . . . 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 19369 . . . . 5 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))
21 riotacl2 7287 . . . . 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 7320 . . . . . . 7 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
2524eqeq2d 2748 . . . . . 6 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2625rexbidv 3172 . . . . 5 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2726elrab 3633 . . . 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 770 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
313ad2antrr 723 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
329ad2antrr 723 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
336ad2antrr 723 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34 simplr 766 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 𝑈))
3512, 17lsmcom2 19327 . . . . . . . . 9 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
361, 7, 19, 35syl3anc 1370 . . . . . . . 8 (𝜑 → (𝑇 𝑈) = (𝑈 𝑇))
3736ad2antrr 723 . . . . . . 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 19368 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑈 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑈 𝑇)) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4031, 32, 33, 38, 39syl31anc 1372 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4111, 12, 16, 17, 1, 7, 18, 19, 13pj1f 19370 . . . . . . . . 9 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4241ad2antrr 723 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4342, 34ffvelcdmd 6999 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
4419ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍𝑈))
4544, 43sseldd 3931 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈))
46 simprl 768 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦𝑈)
4711, 17cntzi 19002 . . . . . . . . 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 2777 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50 oveq2 7321 . . . . . . . 8 (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
5150rspceeqv 3584 . . . . . . 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 764 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54 incom 4145 . . . . . . . . . 10 (𝑈𝑇) = (𝑇𝑈)
5554, 18eqtrid 2789 . . . . . . . . 9 (𝜑 → (𝑈𝑇) = { 0 })
5617, 1, 7, 19cntzrecd 19351 . . . . . . . . 9 (𝜑𝑈 ⊆ (𝑍𝑇))
5711, 12, 16, 17, 7, 1, 55, 56pj1eu 19369 . . . . . . . 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 7320 . . . . . . . . . 10 (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
6059eqeq2d 2748 . . . . . . . . 9 (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
6160rexbidv 3172 . . . . . . . 8 (𝑢 = 𝑦 → (∃𝑣𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣)))
6261riota2 7296 . . . . . . 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 2777 . . . 4 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
6665oveq2d 7329 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
6730, 66eqtr4d 2780 . 2 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
6829, 67rexlimddv 3155 1 ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wrex 3071  ∃!wreu 3348  {crab 3404  cin 3895  wss 3896  {csn 4569  wf 6459  cfv 6463  crio 7269  (class class class)co 7313  Basecbs 16979  +gcplusg 17029  0gc0g 17217  Grpcgrp 18644  SubGrpcsubg 18816  Cntzccntz 18988  LSSumclsm 19306  proj1cpj1 19307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5222  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626  ax-cnex 10997  ax-resscn 10998  ax-1cn 10999  ax-icn 11000  ax-addcl 11001  ax-addrcl 11002  ax-mulcl 11003  ax-mulrcl 11004  ax-mulcom 11005  ax-addass 11006  ax-mulass 11007  ax-distr 11008  ax-i2m1 11009  ax-1ne0 11010  ax-1rid 11011  ax-rnegex 11012  ax-rrecex 11013  ax-cnre 11014  ax-pre-lttri 11015  ax-pre-lttrn 11016  ax-pre-ltadd 11017  ax-pre-mulgt0 11018
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-tr 5203  df-id 5505  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5560  df-we 5562  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-pred 6222  df-ord 6289  df-on 6290  df-lim 6291  df-suc 6292  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-f1 6468  df-fo 6469  df-f1o 6470  df-fv 6471  df-riota 7270  df-ov 7316  df-oprab 7317  df-mpo 7318  df-om 7756  df-1st 7874  df-2nd 7875  df-frecs 8142  df-wrecs 8173  df-recs 8247  df-rdg 8286  df-er 8544  df-en 8780  df-dom 8781  df-sdom 8782  df-pnf 11081  df-mnf 11082  df-xr 11083  df-ltxr 11084  df-le 11085  df-sub 11277  df-neg 11278  df-nn 12044  df-2 12106  df-sets 16932  df-slot 16950  df-ndx 16962  df-base 16980  df-ress 17009  df-plusg 17042  df-0g 17219  df-mgm 18393  df-sgrp 18442  df-mnd 18453  df-grp 18647  df-minusg 18648  df-sbg 18649  df-subg 18819  df-cntz 18990  df-lsm 19308  df-pj1 19309
This theorem is referenced by:  pj1eq  19373  pj1ghm  19376  pj1lmhm  20433
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