Theorem 4atexlemex4 40034

Description: Lemma for 4atexlem7 40036. Show that when 𝐶 = 𝑆, 𝐷 satisfies the existence condition of the consequent. (Contributed by NM, 26-Nov-2012.)

Hypotheses

Ref	Expression
4thatlem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
4thatlem0.l	⊢ ≤ = (le‘𝐾)
4thatlem0.j	⊢ ∨ = (join‘𝐾)
4thatlem0.m	⊢ ∧ = (meet‘𝐾)
4thatlem0.a	⊢ 𝐴 = (Atoms‘𝐾)
4thatlem0.h	⊢ 𝐻 = (LHyp‘𝐾)
4thatlem0.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
4thatlem0.v	⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
4thatlem0.c	⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
4thatlem0.d	⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))

Assertion

Ref	Expression
4atexlemex4	⊢ ((𝜑 ∧ 𝐶 = 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Distinct variable groups:   𝑧,𝐴   𝑧,𝐶   𝑧, ∨   𝑧, ≤   𝑧,𝑃   𝑧,𝑆   𝑧,𝑊   𝑧,𝐷

Allowed substitution hints:   𝜑(𝑧)   𝑄(𝑧)   𝑅(𝑧)   𝑇(𝑧)   𝑈(𝑧)   𝐻(𝑧)   𝐾(𝑧)   ∧ (𝑧)   𝑉(𝑧)

Proof of Theorem 4atexlemex4

Step	Hyp	Ref	Expression
1		4thatlem.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄))))
2		4thatlem0.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		4thatlem0.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		4thatlem0.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5		4thatlem0.u	. . . 4 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
6	1, 2, 3, 4, 5	4atexlemswapqr 40024	. . 3 ⊢ (𝜑 → (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅))))
7		4thatlem0.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
8		4thatlem0.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
9		4thatlem0.v	. . . . 5 ⊢ 𝑉 = ((𝑃 ∨ 𝑆) ∧ 𝑊)
10		4thatlem0.c	. . . . 5 ⊢ 𝐶 = ((𝑄 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
11		4thatlem0.d	. . . . 5 ⊢ 𝐷 = ((𝑅 ∨ 𝑇) ∧ (𝑃 ∨ 𝑆))
12	1, 2, 3, 7, 4, 8, 5, 9, 10, 11	4atexlemcnd 40033	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
13		pm13.18 3012	. . . . . 6 ⊢ ((𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷) → 𝑆 ≠ 𝐷)
14	13	necomd 2986	. . . . 5 ⊢ ((𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷) → 𝐷 ≠ 𝑆)
15	14	expcom 413	. . . 4 ⊢ (𝐶 ≠ 𝐷 → (𝐶 = 𝑆 → 𝐷 ≠ 𝑆))
16	12, 15	syl 17	. . 3 ⊢ (𝜑 → (𝐶 = 𝑆 → 𝐷 ≠ 𝑆))
17		biid 261	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅))) ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅))))
18		eqid 2734	. . . 4 ⊢ ((𝑃 ∨ 𝑅) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊)
19	17, 2, 3, 7, 4, 8, 18, 9, 11	4atexlemex2 40032	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ∧ (𝑃 ∨ 𝑄) = (𝑅 ∨ 𝑄)) ∧ (𝑇 ∈ 𝐴 ∧ (((𝑃 ∨ 𝑅) ∧ 𝑊) ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑅 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑅))) ∧ 𝐷 ≠ 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))
20	6, 16, 19	syl6an 684	. 2 ⊢ (𝜑 → (𝐶 = 𝑆 → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
21	20	imp 406	1 ⊢ ((𝜑 ∧ 𝐶 = 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ≠ wne 2931  ∃wrex 3059   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  lecple 17280  joincjn 18327  meetcmee 18328  Atomscatm 39223  HLchlt 39310  LHypclh 39945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5259  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-proset 18310  df-poset 18329  df-plt 18344  df-lub 18360  df-glb 18361  df-join 18362  df-meet 18363  df-p0 18439  df-p1 18440  df-lat 18446  df-clat 18513  df-oposet 39136  df-ol 39138  df-oml 39139  df-covers 39226  df-ats 39227  df-atl 39258  df-cvlat 39282  df-hlat 39311  df-llines 39459  df-lplanes 39460  df-lhyp 39949

This theorem is referenced by:  4atexlemex6  40035