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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 2737 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
| 5 | pjdm2.k | . . 3 ⊢ 𝐾 = (proj‘𝑊) | |
| 6 | 1, 2, 3, 4, 5 | pjdm 21727 | . 2 ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
| 7 | eqid 2737 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 8 | pjdm2.s | . . . . . 6 ⊢ ⊕ = (LSSum‘𝑊) | |
| 9 | eqid 2737 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 10 | eqid 2737 | . . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
| 11 | phllmod 21648 | . . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
| 12 | 11 | adantr 480 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ LMod) |
| 13 | 2 | lsssssubg 20956 | . . . . . . . 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 3984 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ∈ (SubGrp‘𝑊)) |
| 17 | 1, 2 | lssss 20934 | . . . . . . . 8 ⊢ (𝑇 ∈ 𝐿 → 𝑇 ⊆ 𝑉) |
| 18 | 1, 3, 2 | ocvlss 21690 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ( ⊥ ‘𝑇) ∈ 𝐿) |
| 19 | 17, 18 | sylan2 593 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ 𝐿) |
| 20 | 14, 19 | sseldd 3984 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → ( ⊥ ‘𝑇) ∈ (SubGrp‘𝑊)) |
| 21 | 3, 2, 9 | ocvin 21692 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇 ∩ ( ⊥ ‘𝑇)) = {(0g‘𝑊)}) |
| 22 | lmodabl 20907 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
| 23 | 12, 22 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑊 ∈ Abel) |
| 24 | 10, 23, 16, 20 | ablcntzd 19875 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → 𝑇 ⊆ ((Cntz‘𝑊)‘( ⊥ ‘𝑇))) |
| 25 | 7, 8, 9, 10, 16, 20, 21, 24, 4 | pj1f 19715 | . . . . 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 6753 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ 𝐿) → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉) |
| 28 | fdm 6745 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = (𝑇 ⊕ ( ⊥ ‘𝑇))) | |
| 29 | 28 | eqcomd 2743 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇))) |
| 30 | fdm 6745 | . . . . . . 7 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) = 𝑉) | |
| 31 | 30 | eqeq2d 2748 | . . . . . 6 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → ((𝑇 ⊕ ( ⊥ ‘𝑇)) = dom (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)) ↔ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
| 32 | 29, 31 | syl5ibcom 245 | . . . . 5 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉 → (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)) |
| 33 | feq2 6717 | . . . . . 6 ⊢ ((𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉 → ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 ↔ (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) | |
| 34 | 33 | biimpcd 249 | . . . . 5 ⊢ ((𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):(𝑇 ⊕ ( ⊥ ‘𝑇))⟶𝑉 → ((𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉 → (𝑇(proj1‘𝑊)( ⊥ ‘𝑇)):𝑉⟶𝑉)) |
| 35 | 32, 34 | impbid 212 | . . . 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 283 | 1 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 SubGrpcsubg 19138 Cntzccntz 19333 LSSumclsm 19652 proj1cpj1 19653 Abelcabl 19799 LModclmod 20858 LSubSpclss 20929 PreHilcphl 21642 ocvcocv 21678 projcpj 21720 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-sca 17313 df-vsca 17314 df-ip 17315 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cntz 19335 df-lsm 19654 df-pj1 19655 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-lmod 20860 df-lss 20930 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-phl 21644 df-ocv 21681 df-pj 21723 |
| This theorem is referenced by: pjff 21732 pjf2 21734 pjfo 21735 pjcss 21736 ocvpj 21737 ishil2 21739 pjth2 25474 |
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