Theorem dalem1 39660

Description: Lemma for dath 39737. 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 39635	. 2 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
3	1	dalem-clpjq 39638	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
5	1	dalemkehl 39624	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ HL)
6	1	dalempea 39627	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
7	1	dalemsea 39630	. . . . . . . . . 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 39375	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → 𝑃 ≤ (𝑃 ∨ 𝑆))
12	5, 6, 7, 11	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≤ (𝑃 ∨ 𝑆))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑃 ≤ (𝑃 ∨ 𝑆))
14	1	dalemqea 39628	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
15	1	dalemtea 39631	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
16	8, 9, 10	hlatlej1 39375	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴) → 𝑄 ≤ (𝑄 ∨ 𝑇))
17	5, 14, 15, 16	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ (𝑄 ∨ 𝑇))
18	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑄 ∨ 𝑇))
19		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇))
20	18, 19	breqtrrd 5138	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑃 ∨ 𝑆))
21	1	dalemkelat 39625	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
22	1, 10	dalempeb 39640	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
23	1, 10	dalemqeb 39641	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
24		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 9, 10	hlatjcl 39367	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
26	5, 6, 7, 25	syl3anc 1373	. . . . . . . . . 10 ⊢ (𝜑 → (𝑃 ∨ 𝑆) ∈ (Base‘𝐾))
27	24, 8, 9	latjle12 18416	. . . . . . . . . 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 1374	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
29	28	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ((𝑃 ≤ (𝑃 ∨ 𝑆) ∧ 𝑄 ≤ (𝑃 ∨ 𝑆)) ↔ (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆)))
30	13, 20, 29	mpbi2and 712	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆))
31	1	dalemrea 39629	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
32	1	dalemyeo 39633	. . . . . . . . . 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 39554	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑌 ∈ 𝑂) → 𝑃 ≠ 𝑄)
36	5, 6, 14, 31, 32, 35	syl131anc 1385	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
37	8, 9, 10	ps-1 39478	. . . . . . . . 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 1390	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
39	38	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ((𝑃 ∨ 𝑄) ≤ (𝑃 ∨ 𝑆) ↔ (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆)))
40	30, 39	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑄) = (𝑃 ∨ 𝑆))
41	40	breq2d 5122	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝐶 ≤ (𝑃 ∨ 𝑄) ↔ 𝐶 ≤ (𝑃 ∨ 𝑆)))
42	4, 41	mtbid 324	. . . 4 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆))
43	42	ex 412	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆)))
44	43	necon2ad 2941	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  lecple 17234  joincjn 18279  Latclat 18397  Atomscatm 39263  HLchlt 39350  LPlanesclpl 39493

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 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-proset 18262  df-poset 18281  df-plt 18296  df-lub 18312  df-glb 18313  df-join 18314  df-meet 18315  df-p0 18391  df-lat 18398  df-clat 18465  df-oposet 39176  df-ol 39178  df-oml 39179  df-covers 39266  df-ats 39267  df-atl 39298  df-cvlat 39322  df-hlat 39351  df-llines 39499  df-lplanes 39500

This theorem is referenced by:  dalemcea  39661  dalem2  39662