Theorem dalemqnet 40112

Description: Lemma for dath 40196. Frequently-used utility lemma. (Contributed by NM, 13-Aug-2012.)

Hypotheses

Ref	Expression
dalema.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
dalemc.l	⊢ ≤ = (le‘𝐾)
dalemc.j	⊢ ∨ = (join‘𝐾)
dalemc.a	⊢ 𝐴 = (Atoms‘𝐾)
dalempnes.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalempnes.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)

Assertion

Ref	Expression
dalemqnet	⊢ (𝜑 → 𝑄 ≠ 𝑇)

Proof of Theorem dalemqnet

Step	Hyp	Ref	Expression
1		dalema.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
2	1	dalemkelat 40084	. . 3 ⊢ (𝜑 → 𝐾 ∈ Lat)
3		dalemc.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
4	1, 3	dalemceb 40098	. . 3 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
5	1, 3	dalemteb 40103	. . 3 ⊢ (𝜑 → 𝑇 ∈ (Base‘𝐾))
6	1, 3	dalemueb 40104	. . 3 ⊢ (𝜑 → 𝑈 ∈ (Base‘𝐾))
7		simp322 1326	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))) → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈))
8	1, 7	sylbi 217	. . 3 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑇 ∨ 𝑈))
9		eqid 2737	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
10		dalemc.l	. . . 4 ⊢ ≤ = (le‘𝐾)
11		dalemc.j	. . . 4 ⊢ ∨ = (join‘𝐾)
12	9, 10, 11	latnlej2l 18417	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ 𝑇 ∈ (Base‘𝐾) ∧ 𝑈 ∈ (Base‘𝐾)) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈)) → ¬ 𝐶 ≤ 𝑇)
13	2, 4, 5, 6, 8, 12	syl131anc 1386	. 2 ⊢ (𝜑 → ¬ 𝐶 ≤ 𝑇)
14	1	dalemclqjt 40095	. . . . 5 ⊢ (𝜑 → 𝐶 ≤ (𝑄 ∨ 𝑇))
15		oveq1 7367	. . . . . 6 ⊢ (𝑄 = 𝑇 → (𝑄 ∨ 𝑇) = (𝑇 ∨ 𝑇))
16	15	breq2d 5098	. . . . 5 ⊢ (𝑄 = 𝑇 → (𝐶 ≤ (𝑄 ∨ 𝑇) ↔ 𝐶 ≤ (𝑇 ∨ 𝑇)))
17	14, 16	syl5ibcom 245	. . . 4 ⊢ (𝜑 → (𝑄 = 𝑇 → 𝐶 ≤ (𝑇 ∨ 𝑇)))
18	1	dalemkehl 40083	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
19	1	dalemtea 40090	. . . . . 6 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
20	11, 3	hlatjidm 39829	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑇 ∈ 𝐴) → (𝑇 ∨ 𝑇) = 𝑇)
21	18, 19, 20	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑇 ∨ 𝑇) = 𝑇)
22	21	breq2d 5098	. . . 4 ⊢ (𝜑 → (𝐶 ≤ (𝑇 ∨ 𝑇) ↔ 𝐶 ≤ 𝑇))
23	17, 22	sylibd 239	. . 3 ⊢ (𝜑 → (𝑄 = 𝑇 → 𝐶 ≤ 𝑇))
24	23	necon3bd 2947	. 2 ⊢ (𝜑 → (¬ 𝐶 ≤ 𝑇 → 𝑄 ≠ 𝑇))
25	13, 24	mpd 15	1 ⊢ (𝜑 → 𝑄 ≠ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  lecple 17218  joincjn 18268  Latclat 18388  Atomscatm 39723  HLchlt 39810  LPlanesclpl 39952

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-proset 18251  df-poset 18270  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-lat 18389  df-ats 39727  df-atl 39758  df-cvlat 39782  df-hlat 39811

This theorem is referenced by:  dalemcea  40120  dalem2  40121  dalemdnee  40126