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Mirrors > Home > MPE Home > Th. List > Mathboxes > doch2val2 | Structured version Visualization version GIF version |
Description: Double orthocomplement for DVecH vector space. (Contributed by NM, 26-Jul-2014.) |
Ref | Expression |
---|---|
doch2val2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
doch2val2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
doch2val2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
doch2val2.v | ⊢ 𝑉 = (Base‘𝑈) |
doch2val2.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
doch2val2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
doch2val2.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
doch2val2 | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doch2val2.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | doch2val2.x | . . . 4 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
3 | eqid 2739 | . . . . 5 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
4 | doch2val2.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | doch2val2.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | doch2val2.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | doch2val2.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | doch2val2.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
9 | 3, 4, 5, 6, 7, 8 | dochval2 39012 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) = (𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
10 | 1, 2, 9 | syl2anc 587 | . . 3 ⊢ (𝜑 → ( ⊥ ‘𝑋) = (𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) |
11 | 10 | fveq2d 6681 | . 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ( ⊥ ‘(𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))) |
12 | 1 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
13 | hlop 37022 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ OP) |
15 | ssrab2 3970 | . . . . . . 7 ⊢ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼 | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝜑 → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼) |
17 | 4, 5, 6, 7 | dih1rn 38947 | . . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ ran 𝐼) |
18 | 1, 17 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑉 ∈ ran 𝐼) |
19 | sseq2 3904 | . . . . . . . . 9 ⊢ (𝑧 = 𝑉 → (𝑋 ⊆ 𝑧 ↔ 𝑋 ⊆ 𝑉)) | |
20 | 19 | elrab 3589 | . . . . . . . 8 ⊢ (𝑉 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ↔ (𝑉 ∈ ran 𝐼 ∧ 𝑋 ⊆ 𝑉)) |
21 | 18, 2, 20 | sylanbrc 586 | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
22 | 21 | ne0d 4225 | . . . . . 6 ⊢ (𝜑 → {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅) |
23 | 4, 5 | dihintcl 39004 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ({𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ⊆ ran 𝐼 ∧ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ≠ ∅)) → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) |
24 | 1, 16, 22, 23 | syl12anc 836 | . . . . 5 ⊢ (𝜑 → ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) |
25 | eqid 2739 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
26 | 25, 4, 5 | dihcnvcl 38931 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) |
27 | 1, 24, 26 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) |
28 | 25, 3 | opoccl 36854 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾)) |
29 | 14, 27, 28 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾)) |
30 | 25, 3, 4, 5, 8 | dochvalr2 39022 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∈ (Base‘𝐾)) → ( ⊥ ‘(𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))) |
31 | 1, 29, 30 | syl2anc 587 | . 2 ⊢ (𝜑 → ( ⊥ ‘(𝐼‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))) |
32 | 25, 3 | opococ 36855 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) ∈ (Base‘𝐾)) → ((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) |
33 | 14, 27, 32 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) = (◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) |
34 | 33 | fveq2d 6681 | . . 3 ⊢ (𝜑 → (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = (𝐼‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))) |
35 | 4, 5 | dihcnvid2 38933 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧} ∈ ran 𝐼) → (𝐼‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
36 | 1, 24, 35 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
37 | 34, 36 | eqtrd 2774 | . 2 ⊢ (𝜑 → (𝐼‘((oc‘𝐾)‘((oc‘𝐾)‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
38 | 11, 31, 37 | 3eqtrd 2778 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 {crab 3058 ⊆ wss 3844 ∅c0 4212 ∩ cint 4837 ◡ccnv 5525 ran crn 5527 ‘cfv 6340 Basecbs 16589 occoc 16679 OPcops 36832 HLchlt 37010 LHypclh 37644 DVecHcdvh 38738 DIsoHcdih 38888 ocHcoch 39007 |
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 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-riotaBAD 36613 |
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 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-iin 4885 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-tpos 7924 df-undef 7971 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-n0 11980 df-z 12066 df-uz 12328 df-fz 12985 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-sca 16687 df-vsca 16688 df-0g 16821 df-proset 17657 df-poset 17675 df-plt 17687 df-lub 17703 df-glb 17704 df-join 17705 df-meet 17706 df-p0 17768 df-p1 17769 df-lat 17775 df-clat 17837 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-submnd 18076 df-grp 18225 df-minusg 18226 df-sbg 18227 df-subg 18397 df-cntz 18568 df-lsm 18882 df-cmn 19029 df-abl 19030 df-mgp 19362 df-ur 19374 df-ring 19421 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-invr 19547 df-dvr 19558 df-drng 19626 df-lmod 19758 df-lss 19826 df-lsp 19866 df-lvec 19997 df-lsatoms 36636 df-oposet 36836 df-ol 36838 df-oml 36839 df-covers 36926 df-ats 36927 df-atl 36958 df-cvlat 36982 df-hlat 37011 df-llines 37158 df-lplanes 37159 df-lvols 37160 df-lines 37161 df-psubsp 37163 df-pmap 37164 df-padd 37456 df-lhyp 37648 df-laut 37649 df-ldil 37764 df-ltrn 37765 df-trl 37819 df-tendo 38415 df-edring 38417 df-disoa 38689 df-dvech 38739 df-dib 38799 df-dic 38833 df-dih 38889 df-doch 39008 |
This theorem is referenced by: dochspss 39038 |
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