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Theorem lvoli3 39054
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 483 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ Β¬ 𝑄 ≀ 𝑋)
4 eqidd 2728 . . 3 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))
5 breq2 5154 . . . . . 6 (𝑦 = 𝑋 β†’ (π‘Ÿ ≀ 𝑦 ↔ π‘Ÿ ≀ 𝑋))
65notbid 317 . . . . 5 (𝑦 = 𝑋 β†’ (Β¬ π‘Ÿ ≀ 𝑦 ↔ Β¬ π‘Ÿ ≀ 𝑋))
7 oveq1 7431 . . . . . 6 (𝑦 = 𝑋 β†’ (𝑦 ∨ π‘Ÿ) = (𝑋 ∨ π‘Ÿ))
87eqeq2d 2738 . . . . 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 5153 . . . . . 6 (π‘Ÿ = 𝑄 β†’ (π‘Ÿ ≀ 𝑋 ↔ 𝑄 ≀ 𝑋))
1110notbid 317 . . . . 5 (π‘Ÿ = 𝑄 β†’ (Β¬ π‘Ÿ ≀ 𝑋 ↔ Β¬ 𝑄 ≀ 𝑋))
12 oveq2 7432 . . . . . 6 (π‘Ÿ = 𝑄 β†’ (𝑋 ∨ π‘Ÿ) = (𝑋 ∨ 𝑄))
1312eqeq2d 2738 . . . . 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 3622 . . 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 38840 . . . 4 (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝐾 ∈ Lat)
19 eqid 2727 . . . . . 6 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
20 lvoli3.p . . . . . 6 𝑃 = (LPlanesβ€˜πΎ)
2119, 20lplnbase 39011 . . . . 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 38765 . . . . 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 18436 . . . 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 39053 . . 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 582 . 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 256 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3066   class class class wbr 5150  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  lecple 17245  joincjn 18308  Latclat 18428  Atomscatm 38739  HLchlt 38826  LPlanesclpl 38969  LVolsclvol 38970
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 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-proset 18292  df-poset 18310  df-plt 18327  df-lub 18343  df-glb 18344  df-join 18345  df-meet 18346  df-p0 18422  df-lat 18429  df-clat 18496  df-oposet 38652  df-ol 38654  df-oml 38655  df-covers 38742  df-ats 38743  df-atl 38774  df-cvlat 38798  df-hlat 38827  df-lplanes 38976  df-lvols 38977
This theorem is referenced by:  dalem9  39149  dalem39  39188
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