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Theorem pj1id 18557
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 18043 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . . 6 (𝜑𝐺 ∈ Grp)
4 eqid 2795 . . . . . . . 8 (Base‘𝐺) = (Base‘𝐺)
54subgss 18039 . . . . . . 7 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
61, 5syl 17 . . . . . 6 (𝜑𝑇 ⊆ (Base‘𝐺))
7 pj1eu.3 . . . . . . 7 (𝜑𝑈 ∈ (SubGrp‘𝐺))
84subgss 18039 . . . . . . 7 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
97, 8syl 17 . . . . . 6 (𝜑𝑈 ⊆ (Base‘𝐺))
103, 6, 93jca 1121 . . . . 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 18553 . . . . 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 18554 . . . . 5 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ∃!𝑥𝑇𝑦𝑈 𝑋 = (𝑥 + 𝑦))
21 riotacl2 6995 . . . . 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 2883 . . 3 ((𝜑𝑋 ∈ (𝑇 𝑈)) → ((𝑇𝑃𝑈)‘𝑋) ∈ {𝑥𝑇 ∣ ∃𝑦𝑈 𝑋 = (𝑥 + 𝑦)})
24 oveq1 7028 . . . . . . 7 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑥 + 𝑦) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
2524eqeq2d 2805 . . . . . 6 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (𝑋 = (𝑥 + 𝑦) ↔ 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2625rexbidv 3260 . . . . 5 (𝑥 = ((𝑇𝑃𝑈)‘𝑋) → (∃𝑦𝑈 𝑋 = (𝑥 + 𝑦) ↔ ∃𝑦𝑈 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦)))
2726elrab 3619 . . . 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 769 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
313ad2antrr 722 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝐺 ∈ Grp)
329ad2antrr 722 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑈 ⊆ (Base‘𝐺))
336ad2antrr 722 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (Base‘𝐺))
34 simplr 765 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑇 𝑈))
3512, 17lsmcom2 18515 . . . . . . . . 9 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) = (𝑈 𝑇))
361, 7, 19, 35syl3anc 1364 . . . . . . . 8 (𝜑 → (𝑇 𝑈) = (𝑈 𝑇))
3736ad2antrr 722 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇 𝑈) = (𝑈 𝑇))
3834, 37eleqtrd 2885 . . . . . 6 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 ∈ (𝑈 𝑇))
394, 11, 12, 13pj1val 18553 . . . . . 6 (((𝐺 ∈ Grp ∧ 𝑈 ⊆ (Base‘𝐺) ∧ 𝑇 ⊆ (Base‘𝐺)) ∧ 𝑋 ∈ (𝑈 𝑇)) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4031, 32, 33, 38, 39syl31anc 1366 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)))
4111, 12, 16, 17, 1, 7, 18, 19, 13pj1f 18555 . . . . . . . . 9 (𝜑 → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4241ad2antrr 722 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑇𝑃𝑈):(𝑇 𝑈)⟶𝑇)
4342, 34ffvelrnd 6722 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ 𝑇)
4419ad2antrr 722 . . . . . . . . . 10 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑇 ⊆ (𝑍𝑈))
4544, 43sseldd 3894 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑇𝑃𝑈)‘𝑋) ∈ (𝑍𝑈))
46 simprl 767 . . . . . . . . 9 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑦𝑈)
4711, 17cntzi 18205 . . . . . . . . 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 2831 . . . . . . 7 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
50 oveq2 7029 . . . . . . . 8 (𝑣 = ((𝑇𝑃𝑈)‘𝑋) → (𝑦 + 𝑣) = (𝑦 + ((𝑇𝑃𝑈)‘𝑋)))
5150rspceeqv 3577 . . . . . . 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 763 . . . . . . . 8 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝜑)
54 incom 4103 . . . . . . . . . 10 (𝑈𝑇) = (𝑇𝑈)
5554, 18syl5eq 2843 . . . . . . . . 9 (𝜑 → (𝑈𝑇) = { 0 })
5617, 1, 7, 19cntzrecd 18536 . . . . . . . . 9 (𝜑𝑈 ⊆ (𝑍𝑇))
5711, 12, 16, 17, 7, 1, 55, 56pj1eu 18554 . . . . . . . 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 7028 . . . . . . . . . 10 (𝑢 = 𝑦 → (𝑢 + 𝑣) = (𝑦 + 𝑣))
6059eqeq2d 2805 . . . . . . . . 9 (𝑢 = 𝑦 → (𝑋 = (𝑢 + 𝑣) ↔ 𝑋 = (𝑦 + 𝑣)))
6160rexbidv 3260 . . . . . . . 8 (𝑢 = 𝑦 → (∃𝑣𝑇 𝑋 = (𝑢 + 𝑣) ↔ ∃𝑣𝑇 𝑋 = (𝑦 + 𝑣)))
6261riota2 7004 . . . . . . 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 233 . . . . 5 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (𝑢𝑈𝑣𝑇 𝑋 = (𝑢 + 𝑣)) = 𝑦)
6540, 64eqtrd 2831 . . . 4 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → ((𝑈𝑃𝑇)‘𝑋) = 𝑦)
6665oveq2d 7037 . . 3 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)) = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))
6730, 66eqtr4d 2834 . 2 (((𝜑𝑋 ∈ (𝑇 𝑈)) ∧ (𝑦𝑈𝑋 = (((𝑇𝑃𝑈)‘𝑋) + 𝑦))) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
6829, 67rexlimddv 3254 1 ((𝜑𝑋 ∈ (𝑇 𝑈)) → 𝑋 = (((𝑇𝑃𝑈)‘𝑋) + ((𝑈𝑃𝑇)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wrex 3106  ∃!wreu 3107  {crab 3109  cin 3862  wss 3863  {csn 4476  wf 6226  cfv 6230  crio 6981  (class class class)co 7021  Basecbs 16317  +gcplusg 16399  0gc0g 16547  Grpcgrp 17866  SubGrpcsubg 18032  Cntzccntz 18191  LSSumclsm 18494  proj1cpj1 18495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226  ax-un 7324  ax-cnex 10444  ax-resscn 10445  ax-1cn 10446  ax-icn 10447  ax-addcl 10448  ax-addrcl 10449  ax-mulcl 10450  ax-mulrcl 10451  ax-mulcom 10452  ax-addass 10453  ax-mulass 10454  ax-distr 10455  ax-i2m1 10456  ax-1ne0 10457  ax-1rid 10458  ax-rnegex 10459  ax-rrecex 10460  ax-cnre 10461  ax-pre-lttri 10462  ax-pre-lttrn 10463  ax-pre-ltadd 10464  ax-pre-mulgt0 10465
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-pss 3880  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-tp 4481  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-tr 5069  df-id 5353  df-eprel 5358  df-po 5367  df-so 5368  df-fr 5407  df-we 5409  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-pred 6028  df-ord 6074  df-on 6075  df-lim 6076  df-suc 6077  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-oprab 7025  df-mpo 7026  df-om 7442  df-1st 7550  df-2nd 7551  df-wrecs 7803  df-recs 7865  df-rdg 7903  df-er 8144  df-en 8363  df-dom 8364  df-sdom 8365  df-pnf 10528  df-mnf 10529  df-xr 10530  df-ltxr 10531  df-le 10532  df-sub 10724  df-neg 10725  df-nn 11492  df-2 11553  df-ndx 16320  df-slot 16321  df-base 16323  df-sets 16324  df-ress 16325  df-plusg 16412  df-0g 16549  df-mgm 17686  df-sgrp 17728  df-mnd 17739  df-grp 17869  df-minusg 17870  df-sbg 17871  df-subg 18035  df-cntz 18193  df-lsm 18496  df-pj1 18497
This theorem is referenced by:  pj1eq  18558  pj1ghm  18561  pj1lmhm  19567
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