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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihlatat | Structured version Visualization version GIF version |
Description: The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.) |
Ref | Expression |
---|---|
dihlatat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihlatat.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihlatat.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihlatat.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihlatat.l | ⊢ 𝐿 = (LSAtoms‘𝑈) |
Ref | Expression |
---|---|
dihlatat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐿) → (◡𝐼‘𝑄) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihlatat.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dihlatat.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | id 22 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | dvhlvec 36715 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec) |
5 | eqid 2651 | . . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
6 | eqid 2651 | . . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈) | |
7 | eqid 2651 | . . . . 5 ⊢ (0g‘𝑈) = (0g‘𝑈) | |
8 | dihlatat.l | . . . . 5 ⊢ 𝐿 = (LSAtoms‘𝑈) | |
9 | 5, 6, 7, 8 | islsat 34596 | . . . 4 ⊢ (𝑈 ∈ LVec → (𝑄 ∈ 𝐿 ↔ ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))) |
10 | 4, 9 | syl 17 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑄 ∈ 𝐿 ↔ ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}))) |
11 | 10 | biimpa 500 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐿) → ∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣})) |
12 | eldifsn 4350 | . . . . . 6 ⊢ (𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)}) ↔ (𝑣 ∈ (Base‘𝑈) ∧ 𝑣 ≠ (0g‘𝑈))) | |
13 | dihlatat.a | . . . . . . . 8 ⊢ 𝐴 = (Atoms‘𝐾) | |
14 | dihlatat.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
15 | 13, 1, 2, 5, 7, 6, 14 | dihlspsnat 36939 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑣 ∈ (Base‘𝑈) ∧ 𝑣 ≠ (0g‘𝑈)) → (◡𝐼‘((LSpan‘𝑈)‘{𝑣})) ∈ 𝐴) |
16 | 15 | 3expb 1285 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑣 ∈ (Base‘𝑈) ∧ 𝑣 ≠ (0g‘𝑈))) → (◡𝐼‘((LSpan‘𝑈)‘{𝑣})) ∈ 𝐴) |
17 | 12, 16 | sylan2b 491 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})) → (◡𝐼‘((LSpan‘𝑈)‘{𝑣})) ∈ 𝐴) |
18 | fveq2 6229 | . . . . . 6 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (◡𝐼‘𝑄) = (◡𝐼‘((LSpan‘𝑈)‘{𝑣}))) | |
19 | 18 | eleq1d 2715 | . . . . 5 ⊢ (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → ((◡𝐼‘𝑄) ∈ 𝐴 ↔ (◡𝐼‘((LSpan‘𝑈)‘{𝑣})) ∈ 𝐴)) |
20 | 17, 19 | syl5ibrcom 237 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})) → (𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (◡𝐼‘𝑄) ∈ 𝐴)) |
21 | 20 | rexlimdva 3060 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (◡𝐼‘𝑄) ∈ 𝐴)) |
22 | 21 | adantr 480 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐿) → (∃𝑣 ∈ ((Base‘𝑈) ∖ {(0g‘𝑈)})𝑄 = ((LSpan‘𝑈)‘{𝑣}) → (◡𝐼‘𝑄) ∈ 𝐴)) |
23 | 11, 22 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐿) → (◡𝐼‘𝑄) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∃wrex 2942 ∖ cdif 3604 {csn 4210 ◡ccnv 5142 ‘cfv 5926 Basecbs 15904 0gc0g 16147 LSpanclspn 19019 LVecclvec 19150 LSAtomsclsa 34579 Atomscatm 34868 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 DIsoHcdih 36834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tendo 36360 df-edring 36362 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 |
This theorem is referenced by: dihatexv 36944 dihjat4 37039 dvh4dimat 37044 |
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