Theorem 4atexlemex4 37211

Description: Lemma for 4atexlem7 37213. 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 37201	. . 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 37210	. . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐷)
13		pm13.18 3099	. . . . . 6 ⊢ ((𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷) → 𝑆 ≠ 𝐷)
14	13	necomd 3073	. . . . 5 ⊢ ((𝐶 = 𝑆 ∧ 𝐶 ≠ 𝐷) → 𝐷 ≠ 𝑆)
15	14	expcom 416	. . . 4 ⊢ (𝐶 ≠ 𝐷 → (𝐶 = 𝑆 → 𝐷 ≠ 𝑆))
16	12, 15	syl 17	. . 3 ⊢ (𝜑 → (𝐶 = 𝑆 → 𝐷 ≠ 𝑆))
17		biid 263	. . . 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 2823	. . . 4 ⊢ ((𝑃 ∨ 𝑅) ∧ 𝑊) = ((𝑃 ∨ 𝑅) ∧ 𝑊)
19	17, 2, 3, 7, 4, 8, 18, 9, 11	4atexlemex2 37209	. . 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 682	. 2 ⊢ (𝜑 → (𝐶 = 𝑆 → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧))))
21	20	imp 409	1 ⊢ ((𝜑 ∧ 𝐶 = 𝑆) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑃 ∨ 𝑧) = (𝑆 ∨ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ∃wrex 3141   class class class wbr 5068  ‘cfv 6357  (class class class)co 7158  lecple 16574  joincjn 17556  meetcmee 17557  Atomscatm 36401  HLchlt 36488  LHypclh 37122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-proset 17540  df-poset 17558  df-plt 17570  df-lub 17586  df-glb 17587  df-join 17588  df-meet 17589  df-p0 17651  df-p1 17652  df-lat 17658  df-clat 17720  df-oposet 36314  df-ol 36316  df-oml 36317  df-covers 36404  df-ats 36405  df-atl 36436  df-cvlat 36460  df-hlat 36489  df-llines 36636  df-lplanes 36637  df-lhyp 37126

This theorem is referenced by:  4atexlemex6  37212