Theorem dalem1 38518

Description: Lemma for dath 38595. Show the lines 𝑃𝑆 and 𝑄𝑇 are different. (Contributed by NM, 9-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)
dalem1.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem1.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)

Assertion

Ref	Expression
dalem1	⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))

Proof of Theorem dalem1

Step	Hyp	Ref	Expression
1		dalema.ph	. . 3 ⊢ (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	dalemclpjs 38493	. 2 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
3	1	dalem-clpjq 38496	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
4	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
5	1	dalemkehl 38482	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ HL)
6	1	dalempea 38485	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
7	1	dalemsea 38488	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
8		dalemc.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
9		dalemc.j	. . . . . . . . . . 11 ⊢ ∨ = (join‘𝐾)
10		dalemc.a	. . . . . . . . . . 11 ⊢ 𝐴 = (Atoms‘𝐾)
11	8, 9, 10	hlatlej1 38233	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑆))
12	5, 6, 7, 11	syl3anc 1371	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑆))
13	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑃 ≤ (𝑃 ∨ 𝑆))
14	1	dalemqea 38486	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
15	1	dalemtea 38489	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
16	8, 9, 10	hlatlej1 38233	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑇))
17	5, 14, 15, 16	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑄 ∨ 𝑇))
18	17	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑄 ∨ 𝑇))
19		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇))
20	18, 19	breqtrrd 5175	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑃 ∨ 𝑆))
21	1	dalemkelat 38483	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
22	1, 10	dalempeb 38498	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
23	1, 10	dalemqeb 38499	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
24		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 9, 10	hlatjcl 38225	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
26	5, 6, 7, 25	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
27	24, 8, 9	latjle12 18399	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
28	21, 22, 23, 26, 27	syl13anc 1372	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
29	28	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
30	13, 20, 29	mpbi2and 710	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆))
31	1	dalemrea 38487	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
32	1	dalemyeo 38491	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑂)
33		dalem1.o	. . . . . . . . . . 11 ⊢ 𝑂 = (LPlanes‘𝐾)
34		dalem1.y	. . . . . . . . . . 11 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
35	9, 10, 33, 34	lplnri1 38412	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑌 ∈ 𝑂) → 𝑃 ≠ 𝑄)
36	5, 6, 14, 31, 32, 35	syl131anc 1383	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
37	8, 9, 10	ps-1 38336	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
38	5, 6, 14, 36, 6, 7, 37	syl132anc 1388	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
39	38	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
40	30, 39	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))
41	40	breq2d 5159	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝐶 ≤ (𝑃 ∨ 𝑄) ↔ 𝐶 ≤ (𝑃 ∨ 𝑆)))
42	4, 41	mtbid 323	. . . 4 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆))
43	42	ex 413	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆)))
44	43	necon2ad 2955	. 2 ⊢ (𝜑 → (𝐶 ≤ (𝑃 ∨ 𝑆) → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇)))
45	2, 44	mpd 15	1 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ≠ (𝑄 ∨ 𝑇))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  lecple 17200  joincjn 18260  Latclat 18380  Atomscatm 38121  HLchlt 38208  LPlanesclpl 38351

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18244  df-poset 18262  df-plt 18279  df-lub 18295  df-glb 18296  df-join 18297  df-meet 18298  df-p0 18374  df-lat 18381  df-clat 18448  df-oposet 38034  df-ol 38036  df-oml 38037  df-covers 38124  df-ats 38125  df-atl 38156  df-cvlat 38180  df-hlat 38209  df-llines 38357  df-lplanes 38358

This theorem is referenced by:  dalemcea  38519  dalem2  38520