Theorem lplnnlelln 39532

Description: A lattice plane is not less than or equal to a lattice line. (Contributed by NM, 14-Jul-2012.)

Hypotheses

Ref	Expression
lplnnlelln.l	⊢ ≤ = (le‘𝐾)
lplnnlelln.n	⊢ 𝑁 = (LLines‘𝐾)
lplnnlelln.p	⊢ 𝑃 = (LPlanes‘𝐾)

Assertion

Ref	Expression
lplnnlelln	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)

Proof of Theorem lplnnlelln

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

Step	Hyp	Ref	Expression
1		simp3 1138	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → 𝑌 ∈ 𝑁)
2		eqid 2730	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2730	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2730	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		lplnnlelln.n	. . . . 5 ⊢ 𝑁 = (LLines‘𝐾)
6	2, 3, 4, 5	islln2 39500	. . . 4 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟)))))
7	6	3ad2ant1 1133	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟)))))
8	1, 7	mpbid 232	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))))
9		simp11 1204	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝐾 ∈ HL)
10		simp12 1205	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑋 ∈ 𝑃)
11		simp2l 1200	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑞 ∈ (Atoms‘𝐾))
12		simp2r 1201	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑟 ∈ (Atoms‘𝐾))
13		lplnnlelln.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
14		lplnnlelln.p	. . . . . . . 8 ⊢ 𝑃 = (LPlanes‘𝐾)
15	13, 3, 4, 14	lplnnle2at 39530	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑃 ∧ 𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ≤ (𝑞(join‘𝐾)𝑟))
16	9, 10, 11, 12, 15	syl13anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 ≤ (𝑞(join‘𝐾)𝑟))
17		simp3r 1203	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → 𝑌 = (𝑞(join‘𝐾)𝑟))
18	17	breq2d 5121	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → (𝑋 ≤ 𝑌 ↔ 𝑋 ≤ (𝑞(join‘𝐾)𝑟)))
19	16, 18	mtbird 325	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) ∧ (𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) ∧ (𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 ≤ 𝑌)
20	19	3exp 1119	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → ((𝑞 ∈ (Atoms‘𝐾) ∧ 𝑟 ∈ (Atoms‘𝐾)) → ((𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟)) → ¬ 𝑋 ≤ 𝑌)))
21	20	rexlimdvv 3194	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → (∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟)) → ¬ 𝑋 ≤ 𝑌))
22	21	adantld 490	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑞 ∈ (Atoms‘𝐾)∃𝑟 ∈ (Atoms‘𝐾)(𝑞 ≠ 𝑟 ∧ 𝑌 = (𝑞(join‘𝐾)𝑟))) → ¬ 𝑋 ≤ 𝑌))
23	8, 22	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  lecple 17233  joincjn 18278  Atomscatm 39251  HLchlt 39338  LLinesclln 39480  LPlanesclpl 39481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-proset 18261  df-poset 18280  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 39164  df-ol 39166  df-oml 39167  df-covers 39254  df-ats 39255  df-atl 39286  df-cvlat 39310  df-hlat 39339  df-llines 39487  df-lplanes 39488

This theorem is referenced by:  lplnnelln  39535