Theorem cdleme25b 41160

Description: Transform cdleme24 41158. 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 41159	. 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 2766	. . 3 ⊢ ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
12		eqid 2766	. . 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 41158	. 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 5115	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ≤ 𝑊 ↔ 𝑡 ≤ 𝑊))
15	14	notbid 321	. . . . 5 ⊢ (𝑠 = 𝑡 → (¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑡 ≤ 𝑊))
16		breq1 5115	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑡 ≤ (𝑃 ∨ 𝑄)))
17	16	notbid 321	. . . . 5 ⊢ (𝑠 = 𝑡 → (¬ 𝑠 ≤ (𝑃 ∨ 𝑄) ↔ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)))
18	15, 17	anbi12d 644	. . . 4 ⊢ (𝑠 = 𝑡 → ((¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ (𝑃 ∨ 𝑄)) ↔ (¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄))))
19		oveq1 7423	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑠 ∨ 𝑈) = (𝑡 ∨ 𝑈))
20		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑡 → (𝑃 ∨ 𝑠) = (𝑃 ∨ 𝑡))
21	20	oveq1d 7431	. . . . . . . . . 10 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑠) ∧ 𝑊) = ((𝑃 ∨ 𝑡) ∧ 𝑊))
22	21	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑠 = 𝑡 → (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)) = (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
23	19, 22	oveq12d 7434	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊))) = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))))
24	8, 23	eqtrid 2813	. . . . . . 7 ⊢ (𝑠 = 𝑡 → 𝐹 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))))
25		oveq2 7424	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → (𝑅 ∨ 𝑠) = (𝑅 ∨ 𝑡))
26	25	oveq1d 7431	. . . . . . 7 ⊢ (𝑠 = 𝑡 → ((𝑅 ∨ 𝑠) ∧ 𝑊) = ((𝑅 ∨ 𝑡) ∧ 𝑊))
27	24, 26	oveq12d 7434	. . . . . 6 ⊢ (𝑠 = 𝑡 → (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊)) = (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊)))
28	27	oveq2d 7432	. . . . 5 ⊢ (𝑠 = 𝑡 → ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑠) ∧ 𝑊))) = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))))
29	9, 28	eqtrid 2813	. . . 4 ⊢ (𝑠 = 𝑡 → 𝑁 = ((𝑃 ∨ 𝑄) ∧ (((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊))) ∨ ((𝑅 ∨ 𝑡) ∧ 𝑊))))
30	18, 29	reusv3 5379	. . 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 232	. 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 66	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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092   class class class wbr 5112  ‘cfv 6540  (class class class)co 7416  Basecbs 17279  lecple 17327  joincjn 18377  meetcmee 18378  Atomscatm 40069  HLchlt 40156  LHypclh 40790

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5241  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-iin 4962  df-br 5113  df-opab 5177  df-mpt 5196  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-proset 18360  df-poset 18379  df-plt 18394  df-lub 18410  df-glb 18411  df-join 18412  df-meet 18413  df-p0 18489  df-p1 18490  df-lat 18498  df-clat 18565  df-oposet 39982  df-ol 39984  df-oml 39985  df-covers 40072  df-ats 40073  df-atl 40104  df-cvlat 40128  df-hlat 40157  df-llines 40304  df-lplanes 40305  df-lvols 40306  df-lines 40307  df-psubsp 40309  df-pmap 40310  df-padd 40602  df-lhyp 40794

This theorem is used by:  cdleme25c  41161