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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 20858 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇) |
4 | 3 | frnd 6521 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) ⊆ 𝑇) |
5 | eqid 2821 | . . . . . . . 8 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
6 | eqid 2821 | . . . . . . . 8 ⊢ (proj1‘𝑊) = (proj1‘𝑊) | |
7 | 5, 6, 1 | pjval 20854 | . . . . . . 7 ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
8 | 7 | ad2antlr 725 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → (𝐾‘𝑇) = (𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))) |
9 | 8 | fveq1d 6672 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = ((𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥)) |
10 | eqid 2821 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
11 | eqid 2821 | . . . . . 6 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
12 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
13 | eqid 2821 | . . . . . 6 ⊢ (Cntz‘𝑊) = (Cntz‘𝑊) | |
14 | phllmod 20774 | . . . . . . . . 9 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
15 | 14 | adantr 483 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ LMod) |
16 | eqid 2821 | . . . . . . . . 9 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
17 | 16 | lsssssubg 19730 | . . . . . . . 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 20855 | . . . . . . . 8 ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ (LSubSp‘𝑊) ∧ (𝑇(LSSum‘𝑊)((ocv‘𝑊)‘𝑇)) = 𝑉))) |
20 | 19 | simprbda 501 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (LSubSp‘𝑊)) |
21 | 18, 20 | sseldd 3968 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ∈ (SubGrp‘𝑊)) |
22 | 2, 16 | lssss 19708 | . . . . . . . . 9 ⊢ (𝑇 ∈ (LSubSp‘𝑊) → 𝑇 ⊆ 𝑉) |
23 | 20, 22 | syl 17 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ 𝑉) |
24 | 2, 5, 16 | ocvlss 20816 | . . . . . . . 8 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ⊆ 𝑉) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
25 | 23, 24 | syldan 593 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (LSubSp‘𝑊)) |
26 | 18, 25 | sseldd 3968 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ((ocv‘𝑊)‘𝑇) ∈ (SubGrp‘𝑊)) |
27 | 5, 16, 12 | ocvin 20818 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ (LSubSp‘𝑊)) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
28 | 20, 27 | syldan 593 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝑇 ∩ ((ocv‘𝑊)‘𝑇)) = {(0g‘𝑊)}) |
29 | lmodabl 19681 | . . . . . . . 8 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) | |
30 | 15, 29 | syl 17 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑊 ∈ Abel) |
31 | 13, 30, 21, 26 | ablcntzd 18977 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → 𝑇 ⊆ ((Cntz‘𝑊)‘((ocv‘𝑊)‘𝑇))) |
32 | 10, 11, 12, 13, 21, 26, 28, 31, 6 | pj1lid 18827 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝑇(proj1‘𝑊)((ocv‘𝑊)‘𝑇))‘𝑥) = 𝑥) |
33 | 9, 32 | eqtrd 2856 | . . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) = 𝑥) |
34 | 3 | ffnd 6515 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇) Fn 𝑉) |
35 | 23 | sselda 3967 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ 𝑉) |
36 | fnfvelrn 6848 | . . . . 5 ⊢ (((𝐾‘𝑇) Fn 𝑉 ∧ 𝑥 ∈ 𝑉) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇)) | |
37 | 34, 35, 36 | syl2an2r 683 | . . . 4 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → ((𝐾‘𝑇)‘𝑥) ∈ ran (𝐾‘𝑇)) |
38 | 33, 37 | eqeltrrd 2914 | . . 3 ⊢ (((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) ∧ 𝑥 ∈ 𝑇) → 𝑥 ∈ ran (𝐾‘𝑇)) |
39 | 4, 38 | eqelssd 3988 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ran (𝐾‘𝑇) = 𝑇) |
40 | dffo2 6594 | . 2 ⊢ ((𝐾‘𝑇):𝑉–onto→𝑇 ↔ ((𝐾‘𝑇):𝑉⟶𝑇 ∧ ran (𝐾‘𝑇) = 𝑇)) | |
41 | 3, 39, 40 | sylanbrc 585 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ⊆ wss 3936 {csn 4567 dom cdm 5555 ran crn 5556 Fn wfn 6350 ⟶wf 6351 –onto→wfo 6353 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 0gc0g 16713 SubGrpcsubg 18273 Cntzccntz 18445 LSSumclsm 18759 proj1cpj1 18760 Abelcabl 18907 LModclmod 19634 LSubSpclss 19703 PreHilcphl 20768 ocvcocv 20804 projcpj 20844 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-sca 16581 df-vsca 16582 df-ip 16583 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-ghm 18356 df-cntz 18447 df-lsm 18761 df-pj1 18762 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-lmod 19636 df-lss 19704 df-lmhm 19794 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-phl 20770 df-ocv 20807 df-pj 20847 |
This theorem is referenced by: (None) |
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