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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih1dimb2 | Structured version Visualization version GIF version |
Description: Isomorphism H at an atom under 𝑊. (Contributed by NM, 27-Apr-2014.) |
Ref | Expression |
---|---|
dih1dimb2.b | ⊢ 𝐵 = (Base‘𝐾) |
dih1dimb2.l | ⊢ ≤ = (le‘𝐾) |
dih1dimb2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dih1dimb2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih1dimb2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dih1dimb2.o | ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dih1dimb2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dih1dimb2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dih1dimb2.n | ⊢ 𝑁 = (LSpan‘𝑈) |
Ref | Expression |
---|---|
dih1dimb2 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{〈𝑓, 𝑂〉}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dih1dimb2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
2 | dih1dimb2.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | dih1dimb2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dih1dimb2.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | eqid 2777 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | cdlemf 36712 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) |
7 | simp3 1129 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) | |
8 | simp1rl 1276 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑄 ∈ 𝐴) | |
9 | 7, 8 | eqeltrd 2858 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴) |
10 | simp1l 1211 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
11 | simp2 1128 | . . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ∈ 𝑇) | |
12 | dih1dimb2.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾) | |
13 | 12, 2, 3, 4, 5 | trlnidatb 36326 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴)) |
14 | 10, 11, 13 | syl2anc 579 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴)) |
15 | 9, 14 | mpbird 249 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ≠ ( I ↾ 𝐵)) |
16 | 7 | fveq2d 6450 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝐼‘𝑄)) |
17 | dih1dimb2.o | . . . . . . . 8 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
18 | dih1dimb2.u | . . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
19 | dih1dimb2.i | . . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
20 | dih1dimb2.n | . . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈) | |
21 | 12, 3, 4, 5, 17, 18, 19, 20 | dih1dimb 37389 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{〈𝑓, 𝑂〉})) |
22 | 10, 11, 21 | syl2anc 579 | . . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{〈𝑓, 𝑂〉})) |
23 | 16, 22 | eqtr3d 2815 | . . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘𝑄) = (𝑁‘{〈𝑓, 𝑂〉})) |
24 | 15, 23 | jca 507 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{〈𝑓, 𝑂〉}))) |
25 | 24 | 3expia 1111 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{〈𝑓, 𝑂〉})))) |
26 | 25 | reximdva 3197 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{〈𝑓, 𝑂〉})))) |
27 | 6, 26 | mpd 15 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{〈𝑓, 𝑂〉}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2106 ≠ wne 2968 ∃wrex 3090 {csn 4397 〈cop 4403 class class class wbr 4886 ↦ cmpt 4965 I cid 5260 ↾ cres 5357 ‘cfv 6135 Basecbs 16255 lecple 16345 LSpanclspn 19366 Atomscatm 35412 HLchlt 35499 LHypclh 36133 LTrncltrn 36250 trLctrl 36307 DVecHcdvh 37227 DIsoHcdih 37377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35102 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-oposet 35325 df-ol 35327 df-oml 35328 df-covers 35415 df-ats 35416 df-atl 35447 df-cvlat 35471 df-hlat 35500 df-llines 35647 df-lplanes 35648 df-lvols 35649 df-lines 35650 df-psubsp 35652 df-pmap 35653 df-padd 35945 df-lhyp 36137 df-laut 36138 df-ldil 36253 df-ltrn 36254 df-trl 36308 df-tendo 36904 df-edring 36906 df-disoa 37178 df-dvech 37228 df-dib 37288 df-dih 37378 |
This theorem is referenced by: dihatlat 37483 dihatexv 37487 |
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