Theorem lvoli3 39534

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 1192	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋 ∈ 𝑃)
2		simpl3 1193	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑄 ∈ 𝐴)
3		simpr 484	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ¬ 𝑄 ≤ 𝑋)
4		eqidd 2741	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))
5		breq2 5170	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑟 ≤ 𝑦 ↔ 𝑟 ≤ 𝑋))
6	5	notbid 318	. . . . 5 ⊢ (𝑦 = 𝑋 → (¬ 𝑟 ≤ 𝑦 ↔ ¬ 𝑟 ≤ 𝑋))
7		oveq1 7455	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 ∨ 𝑟) = (𝑋 ∨ 𝑟))
8	7	eqeq2d 2751	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟) ↔ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟)))
9	6, 8	anbi12d 631	. . . 4 ⊢ (𝑦 = 𝑋 → ((¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟)) ↔ (¬ 𝑟 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟))))
10		breq1 5169	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑟 ≤ 𝑋 ↔ 𝑄 ≤ 𝑋))
11	10	notbid 318	. . . . 5 ⊢ (𝑟 = 𝑄 → (¬ 𝑟 ≤ 𝑋 ↔ ¬ 𝑄 ≤ 𝑋))
12		oveq2 7456	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑋 ∨ 𝑟) = (𝑋 ∨ 𝑄))
13	12	eqeq2d 2751	. . . . 5 ⊢ (𝑟 = 𝑄 → ((𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟) ↔ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄)))
14	11, 13	anbi12d 631	. . . 4 ⊢ (𝑟 = 𝑄 → ((¬ 𝑟 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑟)) ↔ (¬ 𝑄 ≤ 𝑋 ∧ (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))))
15	9, 14	rspc2ev 3648	. . 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 1374	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑃 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟)))
17		simpl1 1191	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝐾 ∈ HL)
18	17	hllatd 39320	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝐾 ∈ Lat)
19		eqid 2740	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
20		lvoli3.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
21	19, 20	lplnbase 39491	. . . . 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 39245	. . . . 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 18509	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑄) ∈ (Base‘𝐾))
28	18, 22, 25, 27	syl3anc 1371	. . 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 39533	. . 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 583	. 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 257	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ∃wrex 3076   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  Latclat 18501  Atomscatm 39219  HLchlt 39306  LPlanesclpl 39449  LVolsclvol 39450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-lplanes 39456  df-lvols 39457

This theorem is referenced by:  dalem9  39629  dalem39  39668