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| Mirrors > Home > MPE Home > Th. List > pjf2 | Structured version Visualization version GIF version | ||
| Description: A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) | 
| Ref | Expression | 
|---|---|
| pjf.k | ⊢ 𝐾 = (proj‘𝑊) | 
| pjf.v | ⊢ 𝑉 = (Base‘𝑊) | 
| Ref | Expression | 
|---|---|
| pjf2 | ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 2 | eqid 2736 | . . 3 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
| 3 | eqid 2736 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 4 | eqid 2736 | . . 3 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
| 5 | phllmod 21649 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
| 6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod) | 
| 7 | eqid 2736 | . . . . . 6 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
| 8 | 7 | lsssssubg 20957 | . . . . 5 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) | 
| 9 | 6, 8 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) | 
| 10 | pjf.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
| 11 | eqid 2736 | . . . . . 6 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
| 12 | pjf.k | . . . . . 6 ⊢ 𝐾 = (proj‘𝑊) | |
| 13 | 10, 7, 11, 2, 12 | pjdm2 21732 | . . . . 5 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉))) | 
| 14 | 13 | simprbda 498 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊)) | 
| 15 | 9, 14 | sseldd 3983 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊)) | 
| 16 | 10, 7 | lssss 20935 | . . . . . 6 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉) | 
| 17 | 14, 16 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉) | 
| 18 | 10, 11, 7 | ocvlss 21691 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) | 
| 19 | 17, 18 | syldan 591 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) | 
| 20 | 9, 19 | sseldd 3983 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊)) | 
| 21 | 11, 7, 3 | ocvin 21693 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) | 
| 22 | 14, 21 | syldan 591 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) | 
| 23 | lmodabl 20908 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
| 24 | 6, 23 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel) | 
| 25 | 4, 24, 15, 20 | ablcntzd 19876 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇))) | 
| 26 | eqid 2736 | . . 3 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
| 27 | 1, 2, 3, 4, 15, 20, 22, 25, 26 | pj1f 19716 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇) | 
| 28 | 11, 26, 12 | pjval 21731 | . . . . 5 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) | 
| 29 | 28 | adantl 481 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) | 
| 30 | 29 | eqcomd 2742 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)) = (𝐾‘𝑇)) | 
| 31 | 13 | simplbda 499 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉) | 
| 32 | 30, 31 | feq12d 6723 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇)):(𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇))⟶𝑇 ↔ (𝐾‘𝑇):𝑉⟶𝑇)) | 
| 33 | 27, 32 | mpbid 232 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∩ cin 3949 ⊆ wss 3950 {csn 4625 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 0gc0g 17485 SubGrpcsubg 19139 Cntzccntz 19334 LSSumclsm 19653 proj1cpj1 19654 Abelcabl 19800 LModclmod 20859 LSubSpclss 20930 PreHilcphl 21643 ocvcocv 21679 projcpj 21721 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-minusg 18956 df-sbg 18957 df-subg 19142 df-ghm 19232 df-cntz 19336 df-lsm 19655 df-pj1 19656 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-lmod 20861 df-lss 20931 df-lmhm 21022 df-lvec 21103 df-sra 21173 df-rgmod 21174 df-phl 21645 df-ocv 21682 df-pj 21724 | 
| This theorem is referenced by: pjfo 21736 | 
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