Theorem lvoli3 36873

Description: Condition implying a 3-dim lattice volume. (Contributed by NM, 2-Aug-2012.)

Hypotheses

Ref	Expression
lvoli3.l	⊢ ≤ = (le‘𝐾)
lvoli3.j	⊢ ∨ = (join‘𝐾)
lvoli3.a	⊢ 𝐴 = (Atoms‘𝐾)
lvoli3.p	⊢ 𝑃 = (LPlanes‘𝐾)
lvoli3.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lvoli3	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) ∈ 𝑉)

Proof of Theorem lvoli3

Dummy variables 𝑦 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1189	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋 ∈ 𝑃)
2		simpl3 1190	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑄 ∈ 𝐴)
3		simpr 488	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ¬ 𝑄 ≤ 𝑋)
4		eqidd 2799	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))
5		breq2 5034	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑟 ≤ 𝑦 ↔ 𝑟 ≤ 𝑋))
6	5	notbid 321	. . . . 5 ⊢ (𝑦 = 𝑋 → (¬ 𝑟 ≤ 𝑦 ↔ ¬ 𝑟 ≤ 𝑋))
7		oveq1 7142	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 ∨ 𝑟) = (𝑋 ∨ 𝑟))
8	7	eqeq2d 2809	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟) ↔ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟)))
9	6, 8	anbi12d 633	. . . 4 ⊢ (𝑦 = 𝑋 → ((¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟)) ↔ (¬ 𝑟 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟))))
10		breq1 5033	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑟 ≤ 𝑋 ↔ 𝑄 ≤ 𝑋))
11	10	notbid 321	. . . . 5 ⊢ (𝑟 = 𝑄 → (¬ 𝑟 ≤ 𝑋 ↔ ¬ 𝑄 ≤ 𝑋))
12		oveq2 7143	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑋 ∨ 𝑟) = (𝑋 ∨ 𝑄))
13	12	eqeq2d 2809	. . . . 5 ⊢ (𝑟 = 𝑄 → ((𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟) ↔ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄)))
14	11, 13	anbi12d 633	. . . 4 ⊢ (𝑟 = 𝑄 → ((¬ 𝑟 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟)) ↔ (¬ 𝑄 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))))
15	9, 14	rspc2ev 3583	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴 ∧ (¬ 𝑄 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))) → ∃𝑦 ∈ 𝑃 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟)))
16	1, 2, 3, 4, 15	syl112anc 1371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑃 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟)))
17		simpl1 1188	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝐾 ∈ HL)
18	17	hllatd 36660	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝐾 ∈ Lat)
19		eqid 2798	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
20		lvoli3.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
21	19, 20	lplnbase 36830	. . . . 5 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ (Base‘𝐾))
22	1, 21	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋 ∈ (Base‘𝐾))
23		lvoli3.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
24	19, 23	atbase 36585	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾))
25	2, 24	syl 17	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑄 ∈ (Base‘𝐾))
26		lvoli3.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
27	19, 26	latjcl 17653	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑄) ∈ (Base‘𝐾))
28	18, 22, 25, 27	syl3anc 1368	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) ∈ (Base‘𝐾))
29		lvoli3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
30		lvoli3.v	. . . 4 ⊢ 𝑉 = (LVols‘𝐾)
31	19, 29, 26, 23, 20, 30	islvol3 36872	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑄) ∈ (Base‘𝐾)) → ((𝑋 ∨ 𝑄) ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟))))
32	17, 28, 31	syl2anc 587	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ((𝑋 ∨ 𝑄) ∈ 𝑉 ↔ ∃𝑦 ∈ 𝑃 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟))))
33	16, 32	mpbird 260	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  joincjn 17546  Latclat 17647  Atomscatm 36559  HLchlt 36646  LPlanesclpl 36788  LVolsclvol 36789

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-lplanes 36795  df-lvols 36796

This theorem is referenced by:  dalem9  36968  dalem39  37007