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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 2823 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
5 | pjdm2.k | . . 3 ⊢ 𝐾 = (proj‘𝑊) | |
6 | 1, 2, 3, 4, 5 | pjdm 20853 | . 2 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
7 | eqid 2823 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
8 | pjdm2.s | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
9 | eqid 2823 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
10 | eqid 2823 | . . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
11 | phllmod 20776 | . . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
12 | 11 | adantr 483 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ LMod) |
13 | 2 | lsssssubg 19732 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝐿 ⊆ (SubGrp‘𝑊)) |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝐿 ⊆ (SubGrp‘𝑊)) |
15 | simpr 487 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ 𝐿) | |
16 | 14, 15 | sseldd 3970 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ (SubGrp‘𝑊)) |
17 | 1, 2 | lssss 19710 | . . . . . . . 8 ⊢ (𝑇 ∈ 𝐿 → 𝑇 ⊆ 𝑉) |
18 | 1, 3, 2 | ocvlss 20818 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ( ⊥ ‘𝑇) ∈ 𝐿) |
19 | 17, 18 | sylan2 594 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ 𝐿) |
20 | 14, 19 | sseldd 3970 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ (SubGrp‘𝑊)) |
21 | 3, 2, 9 | ocvin 20820 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇 ∩ ( ⊥ ‘𝑇)) = {(0g‘𝑊)}) |
22 | lmodabl 19683 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
23 | 12, 22 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ Abel) |
24 | 10, 23, 16, 20 | ablcntzd 18979 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ ((Cntz‘𝑊)‘( ⊥ ‘𝑇))) |
25 | 7, 8, 9, 10, 16, 20, 21, 24, 4 | pj1f 18825 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑇) |
26 | 17 | adantl 484 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ 𝑉) |
27 | 25, 26 | fssd 6530 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉) |
28 | fdm 6524 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = (𝑇 ⊕ ( ⊥ ‘𝑇))) | |
29 | 28 | eqcomd 2829 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇))) |
30 | fdm 6524 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = 𝑉) | |
31 | 30 | eqeq2d 2834 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → ((𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) ↔ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
32 | 29, 31 | syl5ibcom 247 | . . . . 5 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
33 | feq2 6498 | . . . . . 6 ⊢ ((𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 ↔ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) | |
34 | 33 | biimpcd 251 | . . . . 5 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉 → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
35 | 32, 34 | impbid 214 | . . . 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 581 | . 2 ⊢ (𝑊 ∈ PreHil → ((𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉) ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
38 | 6, 37 | syl5bb 285 | 1 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 SubGrpcsubg 18275 Cntzccntz 18447 LSSumclsm 18761 proj1cpj1 18762 Abelcabl 18909 LModclmod 19636 LSubSpclss 19705 PreHilcphl 20770 ocvcocv 20806 projcpj 20846 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-sca 16583 df-vsca 16584 df-ip 16585 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-ghm 18358 df-cntz 18449 df-lsm 18763 df-pj1 18764 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-lmod 19638 df-lss 19706 df-lmhm 19796 df-lvec 19877 df-sra 19946 df-rgmod 19947 df-phl 20772 df-ocv 20809 df-pj 20849 |
This theorem is referenced by: pjff 20858 pjf2 20860 pjfo 20861 pjcss 20862 ocvpj 20863 ishil2 20865 pjth2 24045 |
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