Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihord2cN Structured version   Visualization version   GIF version
Theorem dihord2cN 40396
Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dihjust.b 𝐡 = (Baseβ€˜πΎ)
dihjust.l ≀ = (leβ€˜πΎ)
dihjust.j ∨ = (joinβ€˜πΎ)
dihjust.m ∧ = (meetβ€˜πΎ)
dihjust.a 𝐴 = (Atomsβ€˜πΎ)
dihjust.h 𝐻 = (LHypβ€˜πΎ)
dihjust.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
dihjust.J 𝐽 = ((DIsoCβ€˜πΎ)β€˜π‘Š)
dihjust.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dihjust.s βŠ• = (LSSumβ€˜π‘ˆ)
dihord2c.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dihord2c.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dihord2c.o 𝑂 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
Assertion
Ref Expression
dihord2cN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ βŸ¨π‘“, π‘‚βŸ© ∈ (πΌβ€˜(𝑋 ∧ π‘Š)))
Distinct variable groups:   ∨ ,𝑓   ∧ ,𝑓   βŠ• ,𝑓   𝑓,β„Ž,𝐴   𝑓,𝐼   𝑓,𝐽   𝑅,𝑓   𝐡,𝑓,β„Ž   𝑓,𝐻,β„Ž   𝑓,𝐾,β„Ž   ≀ ,𝑓,β„Ž   𝑇,𝑓,β„Ž   𝑓,π‘Š,β„Ž   𝑓,𝑋
Allowed substitution hints:   βŠ• (β„Ž)   𝑅(β„Ž)   π‘ˆ(𝑓,β„Ž)   𝐼(β„Ž)   𝐽(β„Ž)   ∨ (β„Ž)   ∧ (β„Ž)   𝑂(𝑓,β„Ž)   𝑋(β„Ž)
Proof of Theorem dihord2cN
StepHypRef Expression
1 simp3 1137 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š)))
2 eqidd 2732 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ 𝑂 = 𝑂)
3 simp1 1135 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
4 simp1l 1196 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ 𝐾 ∈ HL)
54hllatd 38538 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ 𝐾 ∈ Lat)
6 simp2 1136 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ 𝑋 ∈ 𝐡)
7 simp1r 1197 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ π‘Š ∈ 𝐻)
8 dihjust.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
9 dihjust.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
108, 9lhpbase 39173 . . . . 5 (π‘Š ∈ 𝐻 β†’ π‘Š ∈ 𝐡)
117, 10syl 17 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ π‘Š ∈ 𝐡)
12 dihjust.m . . . . 5 ∧ = (meetβ€˜πΎ)
138, 12latmcl 18398 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Š ∈ 𝐡) β†’ (𝑋 ∧ π‘Š) ∈ 𝐡)
145, 6, 11, 13syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ (𝑋 ∧ π‘Š) ∈ 𝐡)
15 dihjust.l . . . . 5 ≀ = (leβ€˜πΎ)
168, 15, 12latmle2 18423 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Š ∈ 𝐡) β†’ (𝑋 ∧ π‘Š) ≀ π‘Š)
175, 6, 11, 16syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ (𝑋 ∧ π‘Š) ≀ π‘Š)
18 dihord2c.t . . . 4 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
19 dihord2c.r . . . 4 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
20 dihord2c.o . . . 4 𝑂 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
21 dihjust.i . . . 4 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
228, 15, 9, 18, 19, 20, 21dibopelval3 40323 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑋 ∧ π‘Š) ∈ 𝐡 ∧ (𝑋 ∧ π‘Š) ≀ π‘Š)) β†’ (βŸ¨π‘“, π‘‚βŸ© ∈ (πΌβ€˜(𝑋 ∧ π‘Š)) ↔ ((𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š)) ∧ 𝑂 = 𝑂)))
233, 14, 17, 22syl12anc 834 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ (βŸ¨π‘“, π‘‚βŸ© ∈ (πΌβ€˜(𝑋 ∧ π‘Š)) ↔ ((𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š)) ∧ 𝑂 = 𝑂)))
241, 2, 23mpbir2and 710 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑓 ∈ 𝑇 ∧ (π‘…β€˜π‘“) ≀ (𝑋 ∧ π‘Š))) β†’ βŸ¨π‘“, π‘‚βŸ© ∈ (πΌβ€˜(𝑋 ∧ π‘Š)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232   I cid 5574   β†Ύ cres 5679  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  Latclat 18389  LSSumclsm 19544  Atomscatm 38437  HLchlt 38524  LHypclh 39159  LTrncltrn 39276  trLctrl 39333  DVecHcdvh 40253  DIsoBcdib 40313  DIsoCcdic 40347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-lat 18390  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-lhyp 39163  df-disoa 40204  df-dib 40314
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator