Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pjcss | Structured version Visualization version GIF version |
Description: A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
pjcss.k | ⊢ 𝐾 = (proj‘𝑊) |
pjcss.c | ⊢ 𝐶 = (ClSubSp‘𝑊) |
Ref | Expression |
---|---|
pjcss | ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjcss.c | . . . 4 ⊢ 𝐶 = (ClSubSp‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
4 | eqid 2821 | . . . 4 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
5 | simpl 485 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑊 ∈ PreHil) | |
6 | eqid 2821 | . . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
7 | pjcss.k | . . . . . . 7 ⊢ 𝐾 = (proj‘𝑊) | |
8 | 2, 6, 3, 4, 7 | pjdm2 20855 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥(LSSum‘𝑊)((ocv‘𝑊)‘𝑥)) = (Base‘𝑊)))) |
9 | 8 | simprbda 501 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑥 ∈ (LSubSp‘𝑊)) |
10 | 2, 6 | lssss 19708 | . . . . 5 ⊢ (𝑥 ∈ (LSubSp‘𝑊) → 𝑥 ⊆ (Base‘𝑊)) |
11 | 9, 10 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑥 ⊆ (Base‘𝑊)) |
12 | 2, 3 | ocvss 20814 | . . . . 5 ⊢ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑥)) ⊆ (Base‘𝑊) |
13 | 8 | simplbda 502 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → (𝑥(LSSum‘𝑊)((ocv‘𝑊)‘𝑥)) = (Base‘𝑊)) |
14 | 12, 13 | sseqtrrid 4020 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑥)) ⊆ (𝑥(LSSum‘𝑊)((ocv‘𝑊)‘𝑥))) |
15 | 1, 2, 3, 4, 5, 11, 14 | lsmcss 20836 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑥 ∈ 𝐶) |
16 | 15 | ex 415 | . 2 ⊢ (𝑊 ∈ PreHil → (𝑥 ∈ dom 𝐾 → 𝑥 ∈ 𝐶)) |
17 | 16 | ssrdv 3973 | 1 ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 LSSumclsm 18759 LSubSpclss 19703 PreHilcphl 20768 ocvcocv 20804 ClSubSpccss 20805 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-tpos 7892 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-mulr 16579 df-sca 16581 df-vsca 16582 df-ip 16583 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-mhm 17956 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-oppr 19373 df-rnghom 19467 df-staf 19616 df-srng 19617 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-css 20808 df-pj 20847 |
This theorem is referenced by: ocvpj 20861 ishil2 20863 cldcss 24044 hlhil 24046 |
Copyright terms: Public domain | W3C validator |