Theorem lvoli3 37838

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 1191	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑋 ∈ 𝑃)
2		simpl3 1192	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝑄 ∈ 𝐴)
3		simpr 485	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ¬ 𝑄 ≤ 𝑋)
4		eqidd 2737	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) = (𝑋 ∨ 𝑄))
5		breq2 5093	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑟 ≤ 𝑦 ↔ 𝑟 ≤ 𝑋))
6	5	notbid 317	. . . . 5 ⊢ (𝑦 = 𝑋 → (¬ 𝑟 ≤ 𝑦 ↔ ¬ 𝑟 ≤ 𝑋))
7		oveq1 7336	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 ∨ 𝑟) = (𝑋 ∨ 𝑟))
8	7	eqeq2d 2747	. . . . 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 5092	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑟 ≤ 𝑋 ↔ 𝑄 ≤ 𝑋))
11	10	notbid 317	. . . . 5 ⊢ (𝑟 = 𝑄 → (¬ 𝑟 ≤ 𝑋 ↔ ¬ 𝑄 ≤ 𝑋))
12		oveq2 7337	. . . . . 6 ⊢ (𝑟 = 𝑄 → (𝑋 ∨ 𝑟) = (𝑋 ∨ 𝑄))
13	12	eqeq2d 2747	. . . . 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 3581	. . 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 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → ∃𝑦 ∈ 𝑃 ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑦 ∧ (𝑋 ∨ 𝑄) = (𝑦 ∨ 𝑟)))
17		simpl1 1190	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝐾 ∈ HL)
18	17	hllatd 37624	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → 𝐾 ∈ Lat)
19		eqid 2736	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
20		lvoli3.p	. . . . . 6 ⊢ 𝑃 = (LPlanes‘𝐾)
21	19, 20	lplnbase 37795	. . . . 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 37549	. . . . 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 18246	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾)) → (𝑋 ∨ 𝑄) ∈ (Base‘𝐾))
28	18, 22, 25, 27	syl3anc 1370	. . 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 37837	. . 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 584	. 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 256	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ 𝑋) → (𝑋 ∨ 𝑄) ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃wrex 3070   class class class wbr 5089  ‘cfv 6473  (class class class)co 7329  Basecbs 17001  lecple 17058  joincjn 18118  Latclat 18238  Atomscatm 37523  HLchlt 37610  LPlanesclpl 37753  LVolsclvol 37754

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 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-proset 18102  df-poset 18120  df-plt 18137  df-lub 18153  df-glb 18154  df-join 18155  df-meet 18156  df-p0 18232  df-lat 18239  df-clat 18306  df-oposet 37436  df-ol 37438  df-oml 37439  df-covers 37526  df-ats 37527  df-atl 37558  df-cvlat 37582  df-hlat 37611  df-lplanes 37760  df-lvols 37761

This theorem is referenced by:  dalem9  37933  dalem39  37972