Theorem lvolnlelln 39171

Description: A lattice line cannot majorize a lattice volume. (Contributed by NM, 14-Jul-2012.)

Hypotheses

Ref	Expression
lvolnlelln.l	⊢ ≤ = (le‘𝐾)
lvolnlelln.n	⊢ 𝑁 = (LLines‘𝐾)
lvolnlelln.v	⊢ 𝑉 = (LVols‘𝐾)

Assertion

Ref	Expression
lvolnlelln	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)

Proof of Theorem lvolnlelln

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

Step	Hyp	Ref	Expression
1		simp3 1135	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → 𝑌 ∈ 𝑁)
2		eqid 2725	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
3		eqid 2725	. . . . 5 ⊢ (join‘𝐾) = (join‘𝐾)
4		eqid 2725	. . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5		lvolnlelln.n	. . . . 5 ⊢ 𝑁 = (LLines‘𝐾)
6	2, 3, 4, 5	islln2 39098	. . . 4 ⊢ (𝐾 ∈ HL → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)))))
7	6	3ad2ant1 1130	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → (𝑌 ∈ 𝑁 ↔ (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)))))
8	1, 7	mpbid 231	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → (𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))))
9		simp11 1200	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝐾 ∈ HL)
10		simp12 1201	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑋 ∈ 𝑉)
11		simp2l 1196	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑝 ∈ (Atoms‘𝐾))
12		simp2r 1197	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑞 ∈ (Atoms‘𝐾))
13		lvolnlelln.l	. . . . . . . 8 ⊢ ≤ = (le‘𝐾)
14		lvolnlelln.v	. . . . . . . 8 ⊢ 𝑉 = (LVols‘𝐾)
15	13, 3, 4, 14	lvolnle3at 39169	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾))) → ¬ 𝑋 ≤ ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
16	9, 10, 11, 11, 12, 15	syl23anc 1374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ≤ ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
17		simp3r 1199	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = (𝑝(join‘𝐾)𝑞))
18	3, 4	hlatjidm 38955	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑝 ∈ (Atoms‘𝐾)) → (𝑝(join‘𝐾)𝑝) = 𝑝)
19	9, 11, 18	syl2anc 582	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑝(join‘𝐾)𝑝) = 𝑝)
20	19	oveq1d 7433	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞) = (𝑝(join‘𝐾)𝑞))
21	17, 20	eqtr4d 2768	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → 𝑌 = ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞))
22	21	breq2d 5161	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → (𝑋 ≤ 𝑌 ↔ 𝑋 ≤ ((𝑝(join‘𝐾)𝑝)(join‘𝐾)𝑞)))
23	16, 22	mtbird 324	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) ∧ (𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) ∧ (𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ≤ 𝑌)
24	23	3exp 1116	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ((𝑝 ∈ (Atoms‘𝐾) ∧ 𝑞 ∈ (Atoms‘𝐾)) → ((𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 ≤ 𝑌)))
25	24	rexlimdvv 3200	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → (∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞)) → ¬ 𝑋 ≤ 𝑌))
26	25	adantld 489	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ((𝑌 ∈ (Base‘𝐾) ∧ ∃𝑝 ∈ (Atoms‘𝐾)∃𝑞 ∈ (Atoms‘𝐾)(𝑝 ≠ 𝑞 ∧ 𝑌 = (𝑝(join‘𝐾)𝑞))) → ¬ 𝑋 ≤ 𝑌))
27	8, 26	mpd 15	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 ≤ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929  ∃wrex 3059   class class class wbr 5149  ‘cfv 6548  (class class class)co 7418  Basecbs 17181  lecple 17241  joincjn 18304  Atomscatm 38849  HLchlt 38936  LLinesclln 39078  LVolsclvol 39080

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 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7740

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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7374  df-ov 7421  df-oprab 7422  df-proset 18288  df-poset 18306  df-plt 18323  df-lub 18339  df-glb 18340  df-join 18341  df-meet 18342  df-p0 18418  df-lat 18425  df-clat 18492  df-oposet 38762  df-ol 38764  df-oml 38765  df-covers 38852  df-ats 38853  df-atl 38884  df-cvlat 38908  df-hlat 38937  df-llines 39085  df-lplanes 39086  df-lvols 39087

This theorem is referenced by:  lvolnelln  39176

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