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Theorem lvoli3 38959
Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)
Hypotheses
Ref Expression
lvoli3.l ≀ = (leβ€˜πΎ)
lvoli3.j ∨ = (joinβ€˜πΎ)
lvoli3.a 𝐴 = (Atomsβ€˜πΎ)
lvoli3.p 𝑃 = (LPlanesβ€˜πΎ)
lvoli3.v 𝑉 = (LVolsβ€˜πΎ)
Assertion
Ref Expression
lvoli3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑋 ∨ 𝑄) ∈ 𝑉)

Proof of Theorem lvoli3
Dummy variables 𝑦 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1189 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑋 ∈ 𝑃)
2 simpl3 1190 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑄 ∈ 𝐴)
3 simpr 484 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ Β¬ 𝑄 ≀ 𝑋)
4 eqidd 2727 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))
5 breq2 5145 . . . . . 6 (𝑦 = 𝑋 β†’ (π‘Ÿ ≀ 𝑦 ↔ π‘Ÿ ≀ 𝑋))
65notbid 318 . . . . 5 (𝑦 = 𝑋 β†’ (Β¬ π‘Ÿ ≀ 𝑦 ↔ Β¬ π‘Ÿ ≀ 𝑋))
7 oveq1 7411 . . . . . 6 (𝑦 = 𝑋 β†’ (𝑦 ∨ π‘Ÿ) = (𝑋 ∨ π‘Ÿ))
87eqeq2d 2737 . . . . 5 (𝑦 = 𝑋 β†’ ((𝑋 ∨ 𝑄) = (𝑦 ∨ π‘Ÿ) ↔ (𝑋 ∨ 𝑄) = (𝑋 ∨ π‘Ÿ)))
96, 8anbi12d 630 . . . 4 (𝑦 = 𝑋 β†’ ((Β¬ π‘Ÿ ≀ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ π‘Ÿ)) ↔ (Β¬ π‘Ÿ ≀ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ π‘Ÿ))))
10 breq1 5144 . . . . . 6 (π‘Ÿ = 𝑄 β†’ (π‘Ÿ ≀ 𝑋 ↔ 𝑄 ≀ 𝑋))
1110notbid 318 . . . . 5 (π‘Ÿ = 𝑄 β†’ (Β¬ π‘Ÿ ≀ 𝑋 ↔ Β¬ 𝑄 ≀ 𝑋))
12 oveq2 7412 . . . . . 6 (π‘Ÿ = 𝑄 β†’ (𝑋 ∨ π‘Ÿ) = (𝑋 ∨ 𝑄))
1312eqeq2d 2737 . . . . 5 (π‘Ÿ = 𝑄 β†’ ((𝑋 ∨ 𝑄) = (𝑋 ∨ π‘Ÿ) ↔ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄)))
1411, 13anbi12d 630 . . . 4 (π‘Ÿ = 𝑄 β†’ ((Β¬ π‘Ÿ ≀ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ π‘Ÿ)) ↔ (Β¬ 𝑄 ≀ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))))
159, 14rspc2ev 3619 . . 3 ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ (Β¬ 𝑄 ≀ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))) β†’ βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ π‘Ÿ)))
161, 2, 3, 4, 15syl112anc 1371 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ π‘Ÿ)))
17 simpl1 1188 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝐾 ∈ HL)
1817hllatd 38745 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝐾 ∈ Lat)
19 eqid 2726 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
20 lvoli3.p . . . . . 6 𝑃 = (LPlanesβ€˜πΎ)
2119, 20lplnbase 38916 . . . . 5 (𝑋 ∈ 𝑃 β†’ 𝑋 ∈ (Baseβ€˜πΎ))
221, 21syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑋 ∈ (Baseβ€˜πΎ))
23 lvoli3.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
2419, 23atbase 38670 . . . . 5 (𝑄 ∈ 𝐴 β†’ 𝑄 ∈ (Baseβ€˜πΎ))
252, 24syl 17 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑄 ∈ (Baseβ€˜πΎ))
26 lvoli3.j . . . . 5 ∨ = (joinβ€˜πΎ)
2719, 26latjcl 18402 . . . 4 ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Baseβ€˜πΎ) ∧ 𝑄 ∈ (Baseβ€˜πΎ)) β†’ (𝑋 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
2818, 22, 25, 27syl3anc 1368 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑋 ∨ 𝑄) ∈ (Baseβ€˜πΎ))
29 lvoli3.l . . . 4 ≀ = (leβ€˜πΎ)
30 lvoli3.v . . . 4 𝑉 = (LVolsβ€˜πΎ)
3119, 29, 26, 23, 20, 30islvol3 38958 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑄) ∈ (Baseβ€˜πΎ)) β†’ ((𝑋 ∨ 𝑄) ∈ 𝑉 ↔ βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ π‘Ÿ))))
3217, 28, 31syl2anc 583 . 2 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ ((𝑋 ∨ 𝑄) ∈ 𝑉 ↔ βˆƒπ‘¦ ∈ 𝑃 βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ π‘Ÿ))))
3316, 32mpbird 257 1 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑋 ∨ 𝑄) ∈ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  lecple 17211  joincjn 18274  Latclat 18394  Atomscatm 38644  HLchlt 38731  LPlanesclpl 38874  LVolsclvol 38875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-lplanes 38881  df-lvols 38882
This theorem is referenced by:  dalem9  39054  dalem39  39093
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