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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihatexv2 | Structured version Visualization version GIF version |
Description: There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.) |
Ref | Expression |
---|---|
dihatexv2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihatexv2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihatexv2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihatexv2.v | ⊢ 𝑉 = (Base‘𝑈) |
dihatexv2.o | ⊢ 0 = (0g‘𝑈) |
dihatexv2.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihatexv2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihatexv2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
Ref | Expression |
---|---|
dihatexv2 | ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2779 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | dihatexv2.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atbase 35867 | . . 3 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
4 | 3 | anim2i 607 | . 2 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → (𝜑 ∧ 𝑄 ∈ (Base‘𝐾))) |
5 | dihatexv2.k | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | 5 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
7 | eldifi 3994 | . . . . . . 7 ⊢ (𝑥 ∈ (𝑉 ∖ { 0 }) → 𝑥 ∈ 𝑉) | |
8 | dihatexv2.h | . . . . . . . 8 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | dihatexv2.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | dihatexv2.v | . . . . . . . 8 ⊢ 𝑉 = (Base‘𝑈) | |
11 | dihatexv2.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
12 | dihatexv2.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
13 | 8, 9, 10, 11, 12 | dihlsprn 37909 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑉) → (𝑁‘{𝑥}) ∈ ran 𝐼) |
14 | 5, 7, 13 | syl2an 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑁‘{𝑥}) ∈ ran 𝐼) |
15 | 1, 8, 12 | dihcnvcl 37849 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾)) |
16 | 6, 14, 15 | syl2anc 576 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾)) |
17 | eleq1a 2862 | . . . . 5 ⊢ ((◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾) → (𝑄 = (◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾))) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑄 = (◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾))) |
19 | 18 | rexlimdva 3230 | . . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥})) → 𝑄 ∈ (Base‘𝐾))) |
20 | 19 | imdistani 561 | . 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥}))) → (𝜑 ∧ 𝑄 ∈ (Base‘𝐾))) |
21 | dihatexv2.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
22 | 5 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
23 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → 𝑄 ∈ (Base‘𝐾)) | |
24 | 1, 2, 8, 9, 10, 21, 11, 12, 22, 23 | dihatexv 37916 | . . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}))) |
25 | 22 | adantr 473 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
26 | 22, 7, 13 | syl2an 586 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑁‘{𝑥}) ∈ ran 𝐼) |
27 | 8, 12 | dihcnvid2 37851 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘{𝑥}))) = (𝑁‘{𝑥})) |
28 | 25, 26, 27 | syl2anc 576 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐼‘(◡𝐼‘(𝑁‘{𝑥}))) = (𝑁‘{𝑥})) |
29 | 28 | eqeq2d 2789 | . . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝐼‘(◡𝐼‘(𝑁‘{𝑥}))) ↔ (𝐼‘𝑄) = (𝑁‘{𝑥}))) |
30 | simplr 756 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑄 ∈ (Base‘𝐾)) | |
31 | 25, 26, 15 | syl2anc 576 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾)) |
32 | 1, 8, 12 | dih11 37843 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (◡𝐼‘(𝑁‘{𝑥})) ∈ (Base‘𝐾)) → ((𝐼‘𝑄) = (𝐼‘(◡𝐼‘(𝑁‘{𝑥}))) ↔ 𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
33 | 25, 30, 31, 32 | syl3anc 1351 | . . . . 5 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝐼‘(◡𝐼‘(𝑁‘{𝑥}))) ↔ 𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
34 | 29, 33 | bitr3d 273 | . . . 4 ⊢ (((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ 𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
35 | 34 | rexbidva 3242 | . . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
36 | 24, 35 | bitrd 271 | . 2 ⊢ ((𝜑 ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
37 | 4, 20, 36 | pm5.21nd 789 | 1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∃wrex 3090 ∖ cdif 3827 {csn 4441 ◡ccnv 5406 ran crn 5408 ‘cfv 6188 Basecbs 16339 0gc0g 16569 LSpanclspn 19465 Atomscatm 35841 HLchlt 35928 LHypclh 36562 DVecHcdvh 37656 DIsoHcdih 37806 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 ax-riotaBAD 35531 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-iin 4795 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-undef 7742 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-1o 7905 df-oadd 7909 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-fin 8310 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-n0 11708 df-z 11794 df-uz 12059 df-fz 12709 df-struct 16341 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-sca 16437 df-vsca 16438 df-0g 16571 df-proset 17396 df-poset 17414 df-plt 17426 df-lub 17442 df-glb 17443 df-join 17444 df-meet 17445 df-p0 17507 df-p1 17508 df-lat 17514 df-clat 17576 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-submnd 17804 df-grp 17894 df-minusg 17895 df-sbg 17896 df-subg 18060 df-cntz 18218 df-lsm 18522 df-cmn 18668 df-abl 18669 df-mgp 18963 df-ur 18975 df-ring 19022 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 df-drng 19227 df-lmod 19358 df-lss 19426 df-lsp 19466 df-lvec 19597 df-lsatoms 35554 df-oposet 35754 df-ol 35756 df-oml 35757 df-covers 35844 df-ats 35845 df-atl 35876 df-cvlat 35900 df-hlat 35929 df-llines 36076 df-lplanes 36077 df-lvols 36078 df-lines 36079 df-psubsp 36081 df-pmap 36082 df-padd 36374 df-lhyp 36566 df-laut 36567 df-ldil 36682 df-ltrn 36683 df-trl 36737 df-tendo 37333 df-edring 37335 df-disoa 37607 df-dvech 37657 df-dib 37717 df-dic 37751 df-dih 37807 |
This theorem is referenced by: djhcvat42 37993 |
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