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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihord2 | Structured version Visualization version GIF version |
Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: do we need ¬ 𝑋 ≤ 𝑊 and ¬ 𝑌 ≤ 𝑊? (Contributed by NM, 4-Mar-2014.) |
Ref | Expression |
---|---|
dihord2.b | ⊢ 𝐵 = (Base‘𝐾) |
dihord2.l | ⊢ ≤ = (le‘𝐾) |
dihord2.j | ⊢ ∨ = (join‘𝐾) |
dihord2.m | ⊢ ∧ = (meet‘𝐾) |
dihord2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihord2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihord2.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
dihord2.J | ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) |
dihord2.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihord2.s | ⊢ ⊕ = (LSSum‘𝑈) |
Ref | Expression |
---|---|
dihord2 | ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑁 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))) → 𝑋 ≤ 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihord2.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihord2.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | dihord2.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
4 | dihord2.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
5 | dihord2.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
6 | dihord2.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | dihord2.i | . . 3 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
8 | dihord2.J | . . 3 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊) | |
9 | dihord2.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | dihord2.s | . . 3 ⊢ ⊕ = (LSSum‘𝑈) | |
11 | eqid 2818 | . . 3 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
12 | eqid 2818 | . . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊) | |
13 | eqid 2818 | . . 3 ⊢ (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (ℎ ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) | |
14 | eqid 2818 | . . 3 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊) | |
15 | eqid 2818 | . . 3 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊) | |
16 | eqid 2818 | . . 3 ⊢ (+g‘𝑈) = (+g‘𝑈) | |
17 | eqid 2818 | . . 3 ⊢ (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = 𝑁) = (℩ℎ ∈ ((LTrn‘𝐾)‘𝑊)(ℎ‘((oc‘𝐾)‘𝑊)) = 𝑁) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | dihord2pre2 38242 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑁 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))) → (𝑄 ∨ (𝑋 ∧ 𝑊)) ≤ (𝑁 ∨ (𝑌 ∧ 𝑊))) |
19 | simp31 1201 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑁 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))) → (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋) | |
20 | simp32 1202 | . 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑁 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))) → (𝑁 ∨ (𝑌 ∧ 𝑊)) = 𝑌) | |
21 | 18, 19, 20 | 3brtr3d 5088 | 1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ∧ (𝑁 ∨ (𝑌 ∧ 𝑊)) = 𝑌 ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))) → 𝑋 ≤ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 I cid 5452 ↾ cres 5550 ‘cfv 6348 ℩crio 7102 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 lecple 16560 occoc 16561 joincjn 17542 meetcmee 17543 LSSumclsm 18688 Atomscatm 36279 HLchlt 36366 LHypclh 37000 LTrncltrn 37117 trLctrl 37174 TEndoctendo 37768 DVecHcdvh 38094 DIsoBcdib 38154 DIsoCcdic 38188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-riotaBAD 35969 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-undef 7928 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-cntz 18385 df-lsm 18690 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-lmod 19565 df-lss 19633 df-lsp 19673 df-lvec 19804 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-llines 36514 df-lplanes 36515 df-lvols 36516 df-lines 36517 df-psubsp 36519 df-pmap 36520 df-padd 36812 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-tendo 37771 df-edring 37773 df-disoa 38045 df-dvech 38095 df-dib 38155 df-dic 38189 |
This theorem is referenced by: dihord4 38274 |
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