Theorem dalem1 37715

Description: Lemma for dath 37792. 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 37690	. 2 ⊢ (𝜑 → 𝐶 ≤ (𝑃 ∨ 𝑆))
3	1	dalem-clpjq 37693	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
4	3	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑄))
5	1	dalemkehl 37679	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ HL)
6	1	dalempea 37682	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
7	1	dalemsea 37685	. . . . . . . . . 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 37431	. . . . . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑃 ≤ (𝑃 ∨ 𝑆))
14	1	dalemqea 37683	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
15	1	dalemtea 37686	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑇 ∈ 𝐴)
16	8, 9, 10	hlatlej1 37431	. . . . . . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑄 ∨ 𝑇))
19		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇))
20	18, 19	breqtrrd 5109	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇)) → 𝑄 ≤ (𝑃 ∨ 𝑆))
21	1	dalemkelat 37680	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ Lat)
22	1, 10	dalempeb 37695	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾))
23	1, 10	dalemqeb 37696	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾))
24		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐾) = (Base‘𝐾)
25	24, 9, 10	hlatjcl 37423	. . . . . . . . . . 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 18213	. . . . . . . . . 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 482	. . . . . . . 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 37684	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝐴)
32	1	dalemyeo 37688	. . . . . . . . . 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 37609	. . . . . . . . . 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 37533	. . . . . . . . 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 482	. . . . . . 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 5093	. . . . 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 414	. . 3 ⊢ (𝜑 → ((𝑃 ∨ 𝑆) = (𝑄 ∨ 𝑇) → ¬ 𝐶 ≤ (𝑃 ∨ 𝑆)))
44	43	necon2ad 2956	. 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 397   ∧ w3a 1087   = wceq 1539   ∈ wcel 2104   ≠ wne 2941   class class class wbr 5081  ‘cfv 6458  (class class class)co 7307  Basecbs 16957  lecple 17014  joincjn 18074  Latclat 18194  Atomscatm 37319  HLchlt 37406  LPlanesclpl 37548

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-proset 18058  df-poset 18076  df-plt 18093  df-lub 18109  df-glb 18110  df-join 18111  df-meet 18112  df-p0 18188  df-lat 18195  df-clat 18262  df-oposet 37232  df-ol 37234  df-oml 37235  df-covers 37322  df-ats 37323  df-atl 37354  df-cvlat 37378  df-hlat 37407  df-llines 37554  df-lplanes 37555

This theorem is referenced by:  dalemcea  37716  dalem2  37717