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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 2728 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
5 | pjdm2.k | . . 3 ⊢ 𝐾 = (proj‘𝑊) | |
6 | 1, 2, 3, 4, 5 | pjdm 21640 | . 2 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
7 | eqid 2728 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
8 | pjdm2.s | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
9 | eqid 2728 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
10 | eqid 2728 | . . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
11 | phllmod 21561 | . . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
12 | 11 | adantr 480 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ LMod) |
13 | 2 | lsssssubg 20841 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝐿 ⊆ (SubGrp‘𝑊)) |
15 | simpr 484 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ 𝐿) | |
16 | 14, 15 | sseldd 3981 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ (SubGrp‘𝑊)) |
17 | 1, 2 | lssss 20819 | . . . . . . . 8 ⊢ (𝑇 ∈ 𝐿 → 𝑇 ⊆ 𝑉) |
18 | 1, 3, 2 | ocvlss 21603 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ( ⊥ ‘𝑇) ∈ 𝐿) |
19 | 17, 18 | sylan2 592 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ 𝐿) |
20 | 14, 19 | sseldd 3981 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ (SubGrp‘𝑊)) |
21 | 3, 2, 9 | ocvin 21605 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇 ∩ ( ⊥ ‘𝑇)) = {(0g‘𝑊)}) |
22 | lmodabl 20791 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
23 | 12, 22 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ Abel) |
24 | 10, 23, 16, 20 | ablcntzd 19811 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ ((Cntz‘𝑊)‘( ⊥ ‘𝑇))) |
25 | 7, 8, 9, 10, 16, 20, 21, 24, 4 | pj1f 19651 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑇) |
26 | 17 | adantl 481 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ 𝑉) |
27 | 25, 26 | fssd 6740 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉) |
28 | fdm 6731 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = (𝑇 ⊕ ( ⊥ ‘𝑇))) | |
29 | 28 | eqcomd 2734 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇))) |
30 | fdm 6731 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = 𝑉) | |
31 | 30 | eqeq2d 2739 | . . . . . 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 6704 | . . . . . 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 578 | . 2 ⊢ (𝑊 ∈ PreHil → ((𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉) ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
38 | 6, 37 | bitrid 283 | 1 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 dom cdm 5678 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 Basecbs 17179 +gcplusg 17232 0gc0g 17420 SubGrpcsubg 19074 Cntzccntz 19265 LSSumclsm 19588 proj1cpj1 19589 Abelcabl 19735 LModclmod 20742 LSubSpclss 20814 PreHilcphl 21555 ocvcocv 21591 projcpj 21633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-sca 17248 df-vsca 17249 df-ip 17250 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-minusg 18893 df-sbg 18894 df-subg 19077 df-ghm 19167 df-cntz 19267 df-lsm 19590 df-pj1 19591 df-cmn 19736 df-abl 19737 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-lmod 20744 df-lss 20815 df-lmhm 20906 df-lvec 20987 df-sra 21057 df-rgmod 21058 df-phl 21557 df-ocv 21594 df-pj 21636 |
This theorem is referenced by: pjff 21645 pjf2 21647 pjfo 21648 pjcss 21649 ocvpj 21650 ishil2 21652 pjth2 25367 |
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