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Mirrors > Home > MPE Home > Th. List > pjdm2 | Structured version Visualization version GIF version |
Description: A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjdm2.v | ⊢ 𝑉 = (Base‘𝑊) |
pjdm2.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
pjdm2.o | ⊢ ⊥ = (ocv‘𝑊) |
pjdm2.s | ⊢ ⊕ = (LSSum‘𝑊) |
pjdm2.k | ⊢ 𝐾 = (proj‘𝑊) |
Ref | Expression |
---|---|
pjdm2 | ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjdm2.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | pjdm2.l | . . 3 ⊢ 𝐿 = (LSubSp‘𝑊) | |
3 | pjdm2.o | . . 3 ⊢ ⊥ = (ocv‘𝑊) | |
4 | eqid 2736 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
5 | pjdm2.k | . . 3 ⊢ 𝐾 = (proj‘𝑊) | |
6 | 1, 2, 3, 4, 5 | pjdm 21113 | . 2 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
7 | eqid 2736 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
8 | pjdm2.s | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
9 | eqid 2736 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
10 | eqid 2736 | . . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
11 | phllmod 21034 | . . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
12 | 11 | adantr 481 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ LMod) |
13 | 2 | lsssssubg 20419 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝐿 ⊆ (SubGrp‘𝑊)) |
15 | simpr 485 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ 𝐿) | |
16 | 14, 15 | sseldd 3945 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ (SubGrp‘𝑊)) |
17 | 1, 2 | lssss 20397 | . . . . . . . 8 ⊢ (𝑇 ∈ 𝐿 → 𝑇 ⊆ 𝑉) |
18 | 1, 3, 2 | ocvlss 21076 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ( ⊥ ‘𝑇) ∈ 𝐿) |
19 | 17, 18 | sylan2 593 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ 𝐿) |
20 | 14, 19 | sseldd 3945 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ (SubGrp‘𝑊)) |
21 | 3, 2, 9 | ocvin 21078 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇 ∩ ( ⊥ ‘𝑇)) = {(0g‘𝑊)}) |
22 | lmodabl 20369 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
23 | 12, 22 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ Abel) |
24 | 10, 23, 16, 20 | ablcntzd 19635 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ ((Cntz‘𝑊)‘( ⊥ ‘𝑇))) |
25 | 7, 8, 9, 10, 16, 20, 21, 24, 4 | pj1f 19479 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑇) |
26 | 17 | adantl 482 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ 𝑉) |
27 | 25, 26 | fssd 6686 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉) |
28 | fdm 6677 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = (𝑇 ⊕ ( ⊥ ‘𝑇))) | |
29 | 28 | eqcomd 2742 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇))) |
30 | fdm 6677 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = 𝑉) | |
31 | 30 | eqeq2d 2747 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → ((𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) ↔ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
32 | 29, 31 | syl5ibcom 244 | . . . . 5 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
33 | feq2 6650 | . . . . . 6 ⊢ ((𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 ↔ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) | |
34 | 33 | biimpcd 248 | . . . . 5 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉 → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
35 | 32, 34 | impbid 211 | . . . 4 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 ↔ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
36 | 27, 35 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 ↔ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
37 | 36 | pm5.32da 579 | . 2 ⊢ (𝑊 ∈ PreHil → ((𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉) ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
38 | 6, 37 | bitrid 282 | 1 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 dom cdm 5633 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +gcplusg 17133 0gc0g 17321 SubGrpcsubg 18922 Cntzccntz 19095 LSSumclsm 19416 proj1cpj1 19417 Abelcabl 19563 LModclmod 20322 LSubSpclss 20392 PreHilcphl 21028 ocvcocv 21064 projcpj 21106 |
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-int 4908 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-map 8767 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-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-sca 17149 df-vsca 17150 df-ip 17151 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-ghm 19006 df-cntz 19097 df-lsm 19418 df-pj1 19419 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-lmod 20324 df-lss 20393 df-lmhm 20483 df-lvec 20564 df-sra 20633 df-rgmod 20634 df-phl 21030 df-ocv 21067 df-pj 21109 |
This theorem is referenced by: pjff 21118 pjf2 21120 pjfo 21121 pjcss 21122 ocvpj 21123 ishil2 21125 pjth2 24804 |
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