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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihord | Structured version Visualization version GIF version |
Description: The isomorphism H is order-preserving. Part of proof after Lemma N of [Crawley] p. 122 line 6. (Contributed by NM, 7-Mar-2014.) |
Ref | Expression |
---|---|
dihord.b | ⊢ 𝐵 = (Base‘𝐾) |
dihord.l | ⊢ ≤ = (le‘𝐾) |
dihord.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihord.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihord | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1246 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simpl2 1248 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵) | |
3 | simprl 787 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ≤ 𝑊) | |
4 | simpl3 1250 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵) | |
5 | simprr 789 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ≤ 𝑊) | |
6 | dihord.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
7 | dihord.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
8 | dihord.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | dihord.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
10 | 6, 7, 8, 9 | dihord3 37331 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
11 | 1, 2, 3, 4, 5, 10 | syl122anc 1502 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
12 | simpl1 1246 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | simpl2 1248 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵) | |
14 | simprl 787 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → 𝑋 ≤ 𝑊) | |
15 | simpl3 1250 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵) | |
16 | simprr 789 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → ¬ 𝑌 ≤ 𝑊) | |
17 | 6, 7, 8, 9 | dihord5a 37337 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌) |
18 | 6, 7, 8, 9 | dihord5b 37333 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) |
19 | 17, 18 | impbida 835 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
20 | 12, 13, 14, 15, 16, 19 | syl122anc 1502 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
21 | simpl1 1246 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
22 | simpl2 1248 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵) | |
23 | simprl 787 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → ¬ 𝑋 ≤ 𝑊) | |
24 | simpl3 1250 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵) | |
25 | simprr 789 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → 𝑌 ≤ 𝑊) | |
26 | 6, 7, 8, 9 | dihord6a 37335 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) → 𝑋 ≤ 𝑌) |
27 | 6, 7, 8, 9 | dihord6b 37334 | . . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)) |
28 | 26, 27 | impbida 835 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
29 | 21, 22, 23, 24, 25, 28 | syl122anc 1502 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
30 | simpl1 1246 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
31 | simpl2 1248 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → 𝑋 ∈ 𝐵) | |
32 | simprl 787 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → ¬ 𝑋 ≤ 𝑊) | |
33 | simpl3 1250 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → 𝑌 ∈ 𝐵) | |
34 | simprr 789 | . . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → ¬ 𝑌 ≤ 𝑊) | |
35 | 6, 7, 8, 9 | dihord4 37332 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
36 | 30, 31, 32, 33, 34, 35 | syl122anc 1502 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑋 ≤ 𝑊 ∧ ¬ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
37 | 11, 20, 29, 36 | 4casesdan 1068 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 class class class wbr 4875 ‘cfv 6127 Basecbs 16229 lecple 16319 HLchlt 35424 LHypclh 36058 DIsoHcdih 37302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tendo 36829 df-edring 36831 df-disoa 37103 df-dvech 37153 df-dib 37213 df-dic 37247 df-dih 37303 |
This theorem is referenced by: dih11 37339 dihcnvord 37348 dihmeetlem1N 37364 dihglblem5apreN 37365 dihglblem5aN 37366 dihglblem4 37371 dihmeetlem9N 37389 dihmeetlem11N 37391 dihlspsnat 37407 dihglblem6 37414 dochvalr 37431 dochss 37439 dvh4dimat 37512 |
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