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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 2738 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
3 | eqid 2738 | . . . 4 ⊢ (ocv‘𝑊) = (ocv‘𝑊) | |
4 | eqid 2738 | . . . 4 ⊢ (LSSum‘𝑊) = (LSSum‘𝑊) | |
5 | simpl 486 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑊 ∈ PreHil) | |
6 | eqid 2738 | . . . . . . 7 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
7 | pjcss.k | . . . . . . 7 ⊢ 𝐾 = (proj‘𝑊) | |
8 | 2, 6, 3, 4, 7 | pjdm2 20527 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → (𝑥 ∈ dom 𝐾 ↔ (𝑥 ∈ (LSubSp‘𝑊) ∧ (𝑥(LSSum‘𝑊)((ocv‘𝑊)‘𝑥)) = (Base‘𝑊)))) |
9 | 8 | simprbda 502 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑥 ∈ (LSubSp‘𝑊)) |
10 | 2, 6 | lssss 19827 | . . . . 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 20486 | . . . . 5 ⊢ ((ocv‘𝑊)‘((ocv‘𝑊)‘𝑥)) ⊆ (Base‘𝑊) |
13 | 8 | simplbda 503 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → (𝑥(LSSum‘𝑊)((ocv‘𝑊)‘𝑥)) = (Base‘𝑊)) |
14 | 12, 13 | sseqtrrid 3930 | . . . 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 20508 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ dom 𝐾) → 𝑥 ∈ 𝐶) |
16 | 15 | ex 416 | . 2 ⊢ (𝑊 ∈ PreHil → (𝑥 ∈ dom 𝐾 → 𝑥 ∈ 𝐶)) |
17 | 16 | ssrdv 3883 | 1 ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 dom cdm 5525 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 LSSumclsm 18877 LSubSpclss 19822 PreHilcphl 20440 ocvcocv 20476 ClSubSpccss 20477 projcpj 20516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-tpos 7921 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-ip 16686 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-mhm 18072 df-grp 18222 df-minusg 18223 df-sbg 18224 df-subg 18394 df-ghm 18474 df-cntz 18565 df-lsm 18879 df-pj1 18880 df-cmn 19026 df-abl 19027 df-mgp 19359 df-ur 19371 df-ring 19418 df-oppr 19495 df-rnghom 19589 df-staf 19735 df-srng 19736 df-lmod 19755 df-lss 19823 df-lmhm 19913 df-lvec 19994 df-sra 20063 df-rgmod 20064 df-phl 20442 df-ocv 20479 df-css 20480 df-pj 20519 |
This theorem is referenced by: ocvpj 20533 ishil2 20535 cldcss 24193 hlhil 24195 |
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