Theorem cdleme25b 39220

Description: Transform cdleme24 39218. TODO get rid of $d's on 𝑈, 𝑁 (Contributed by NM, 1-Jan-2013.)

Hypotheses

Ref	Expression
cdleme24.b	⊢ 𝐵 = (Base‘𝐾)
cdleme24.l	⊢ ≤ = (le‘𝐾)
cdleme24.j	⊢ ∨ = (join‘𝐾)
cdleme24.m	⊢ ∧ = (meet‘𝐾)
cdleme24.a	⊢ 𝐴 = (Atoms‘𝐾)
cdleme24.h	⊢ 𝐻 = (LHyp‘𝐾)
cdleme24.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdleme24.f	⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme24.n	⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊)))

Assertion

Ref	Expression
cdleme25b	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))

Distinct variable groups:   𝑢,𝑠,𝐴   𝐵,𝑠,𝑢   𝐻,𝑠   ∨ ,𝑠,𝑢   𝐾,𝑠   ≤ ,𝑠,𝑢   ∧ ,𝑠,𝑢   𝑃,𝑠,𝑢   𝑄,𝑠,𝑢   𝑅,𝑠,𝑢   𝑊,𝑠,𝑢   𝑢,𝑁   𝑈,𝑠,𝑢

Allowed substitution hints:   𝐹(𝑢,𝑠)   𝐻(𝑢)   𝐾(𝑢)   𝑁(𝑠)

Proof of Theorem cdleme25b

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdleme24.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdleme24.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdleme24.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdleme24.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdleme24.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdleme24.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdleme24.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdleme24.f	. . 3 ⊢ 𝐹 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
9		cdleme24.n	. . 3 ⊢ 𝑁 = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	cdleme25a 39219	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵))
11		eqid 2732	. . 3 ⊢ ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
12		eqid 2732	. . 3 ⊢ ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12	cdleme24 39218	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))))
14		breq1 5151	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ≤ 𝑊 ↔ 𝑡 ≤ 𝑊))
15	14	notbid 317	. . . . 5 ⊢ (𝑠 = 𝑡 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑡 ≤ 𝑊))
16		breq1 5151	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑡 ≤ (𝑃 ∨ 𝑄)))
17	16	notbid 317	. . . . 5 ⊢ (𝑠 = 𝑡 → (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))
18	15, 17	anbi12d 631	. . . 4 ⊢ (𝑠 = 𝑡 → ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ↔ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))))
19		oveq1 7415	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑈) = (𝑡 ∨ 𝑈))
20		oveq2 7416	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑡))
21	20	oveq1d 7423	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑡) ∧ 𝑊))
22	21	oveq2d 7424	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
23	19, 22	oveq12d 7426	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))))
24	8, 23	eqtrid 2784	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝐹 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))))
25		oveq2 7416	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑅 ∨ 𝑠) = (𝑅 ∨ 𝑡))
26	25	oveq1d 7423	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ((𝑅 ∨ 𝑠) ∧ 𝑊) = ((𝑅 ∨ 𝑡) ∧ 𝑊))
27	24, 26	oveq12d 7426	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊)) = (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
28	27	oveq2d 7424	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))))
29	9, 28	eqtrid 2784	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))))
30	18, 29	reusv3 5403	. . 3 ⊢ (∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵) → (∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))) ↔ ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)))
31	30	biimpd 228	. 2 ⊢ (∃𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ 𝑁 ∈ 𝐵) → (∀𝑠 ∈ 𝐴 ∀𝑡 ∈ 𝐴 (((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ∧ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))) → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁)))
32	10, 13, 31	sylc 65	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑃 ≠ 𝑄 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ∃𝑢 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) → 𝑢 = 𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070   class class class wbr 5148  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  lecple 17203  joincjn 18263  meetcmee 18264  Atomscatm 38128  HLchlt 38215  LHypclh 38850

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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854

This theorem is referenced by:  cdleme25c  39221