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Mirrors > Home > MPE Home > Th. List > pjfo | Structured version Visualization version GIF version |
Description: A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjf.k | ⊢ 𝐾 = (proj‘𝑊) |
pjf.v | ⊢ 𝑉 = (Base‘𝑊) |
Ref | Expression |
---|---|
pjfo | ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjf.k | . . 3 ⊢ 𝐾 = (proj‘𝑊) | |
2 | pjf.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
3 | 1, 2 | pjf2 21577 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) |
4 | 3 | frnd 6715 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) ⊆ 𝑇) |
5 | eqid 2724 | . . . . . . . 8 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
6 | eqid 2724 | . . . . . . . 8 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
7 | 5, 6, 1 | pjval 21573 | . . . . . . 7 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
8 | 7 | ad2antlr 724 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
9 | 8 | fveq1d 6883 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = ((𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥)) |
10 | eqid 2724 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2724 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
12 | eqid 2724 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
13 | eqid 2724 | . . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
14 | phllmod 21491 | . . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
15 | 14 | adantr 480 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod) |
16 | eqid 2724 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
17 | 16 | lsssssubg 20795 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
18 | 15, 17 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (LSubSp‘𝑊) ⊆ (SubGrp‘𝑊)) |
19 | 2, 16, 5, 11, 1 | pjdm2 21574 | . . . . . . . 8 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉))) |
20 | 19 | simprbda 498 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊)) |
21 | 18, 20 | sseldd 3975 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊)) |
22 | 2, 16 | lssss 20773 | . . . . . . . . 9 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉) |
23 | 20, 22 | syl 17 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉) |
24 | 2, 5, 16 | ocvlss 21533 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
25 | 23, 24 | syldan 590 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
26 | 18, 25 | sseldd 3975 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊)) |
27 | 5, 16, 12 | ocvin 21535 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
28 | 20, 27 | syldan 590 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
29 | lmodabl 20745 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
30 | 15, 29 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel) |
31 | 13, 30, 21, 26 | ablcntzd 19767 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇))) |
32 | 10, 11, 12, 13, 21, 26, 28, 31, 6 | pj1lid 19611 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥) = 𝑥) |
33 | 9, 32 | eqtrd 2764 | . . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = 𝑥) |
34 | 3 | ffnd 6708 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) Fn 𝑉) |
35 | 23 | sselda 3974 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑉) |
36 | fnfvelrn 7072 | . . . . 5 ⊢ (((𝐾‘𝑇) Fn 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇)) | |
37 | 34, 35, 36 | syl2an2r 682 | . . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇)) |
38 | 33, 37 | eqeltrrd 2826 | . . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ ran (𝐾‘𝑇)) |
39 | 4, 38 | eqelssd 3995 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) = 𝑇) |
40 | dffo2 6799 | . 2 ⊢ ((𝐾‘𝑇):𝑉–onto→𝑇 ↔ ((𝐾‘𝑇):𝑉⟶𝑇 ∧ ran (𝐾‘𝑇) = 𝑇)) | |
41 | 3, 39, 40 | sylanbrc 582 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 ⊆ wss 3940 {csn 4620 dom cdm 5666 ran crn 5667 Fn wfn 6528 ⟶wf 6529 –onto→wfo 6531 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 +gcplusg 17196 0gc0g 17384 SubGrpcsubg 19037 Cntzccntz 19221 LSSumclsm 19544 proj1cpj1 19545 Abelcabl 19691 LModclmod 20696 LSubSpclss 20768 PreHilcphl 21485 ocvcocv 21521 projcpj 21563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-sca 17212 df-vsca 17213 df-ip 17214 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18704 df-grp 18856 df-minusg 18857 df-sbg 18858 df-subg 19040 df-ghm 19129 df-cntz 19223 df-lsm 19546 df-pj1 19547 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-lmod 20698 df-lss 20769 df-lmhm 20860 df-lvec 20941 df-sra 21011 df-rgmod 21012 df-phl 21487 df-ocv 21524 df-pj 21566 |
This theorem is referenced by: (None) |
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