Theorem dalemswapyzps 40893

Description: Lemma for dath 40939. Swap the 𝑌 and 𝑍 planes, along with dummy concurrency (center of perspectivity) atoms 𝑐 and 𝑑, to allow reuse of analogous proofs. (Contributed by NM, 17-Aug-2012.)

Hypotheses

Ref	Expression
dalem.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)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))

Assertion

Ref	Expression
dalemswapyzps	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))))

Proof of Theorem dalemswapyzps

Step	Hyp	Ref	Expression
1		dalem.ps	. . . . 5 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
2	1	dalemddea 40887	. . . 4 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
3	1	dalemccea 40886	. . . 4 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
4	2, 3	jca 521	. . 3 ⊢ (𝜓 → (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴))
5	4	3ad2ant3 1153	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴))
6	1	dalem-ddly 40889	. . . 4 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌)
7	6	3ad2ant3 1153	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌)
8		simp2 1155	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 = 𝑍)
9	8	breq2d 5118	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ≤ 𝑌 ↔ 𝑑 ≤ 𝑍))
10	7, 9	mtbid 327	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑍)
11	1	dalemccnedd 40890	. . . 4 ⊢ (𝜓 → 𝑐 ≠ 𝑑)
12	11	3ad2ant3 1153	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≠ 𝑑)
13	1	dalem-ccly 40888	. . . . 5 ⊢ (𝜓 → ¬ 𝑐 ≤ 𝑌)
14	13	3ad2ant3 1153	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑌)
15	8	breq2d 5118	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≤ 𝑌 ↔ 𝑐 ≤ 𝑍))
16	14, 15	mtbid 327	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ 𝑍)
17	1	dalemclccjdd 40891	. . . . 5 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
18	17	3ad2ant3 1153	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
19		dalem.ph	. . . . . . 7 ⊢ (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)))))
20	19	dalemkehl 40826	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL)
21	20	3ad2ant1 1151	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
22	3	3ad2ant3 1153	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
23	2	3ad2ant3 1153	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
24		dalem.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
25		dalem.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
26	24, 25	hlatjcom 40571	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → (𝑐 ∨ 𝑑) = (𝑑 ∨ 𝑐))
27	21, 22, 23, 26	syl3anc 1398	. . . 4 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) = (𝑑 ∨ 𝑐))
28	18, 27	breqtrd 5134	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ (𝑑 ∨ 𝑐))
29	12, 16, 28	3jca 1146	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐)))
30	5, 10, 29	3jca 1146	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑑 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ ¬ 𝑑 ≤ 𝑍 ∧ (𝑐 ≠ 𝑑 ∧ ¬ 𝑐 ≤ 𝑍 ∧ 𝐶 ≤ (𝑑 ∨ 𝑐))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2959   class class class wbr 5106  ‘cfv 6547  (class class class)co 7430  Basecbs 17402  lecple 17450  joincjn 18500  Atomscatm 40466  HLchlt 40553

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-lub 18533  df-join 18535  df-lat 18621  df-ats 40470  df-atl 40501  df-cvlat 40525  df-hlat 40554

This theorem is used by:  dalem56  40931