Theorem cdlema2N 39837

Description: A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlema2.b	⊢ 𝐵 = (Base‘𝐾)
cdlema2.l	⊢ ≤ = (le‘𝐾)
cdlema2.j	⊢ ∨ = (join‘𝐾)
cdlema2.m	⊢ ∧ = (meet‘𝐾)
cdlema2.z	⊢ 0 = (0.‘𝐾)
cdlema2.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cdlema2N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (𝑅 ∧ 𝑋) = 0 )

Proof of Theorem cdlema2N

Step	Hyp	Ref	Expression
1		simp3ll 1245	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝑅 ≠ 𝑃)
2		simp3rl 1247	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝑃 ≤ 𝑋)
3		simp3rr 1248	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → ¬ 𝑄 ≤ 𝑋)
4		simp3lr 1246	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝑅 ≤ (𝑃 ∨ 𝑄))
5	2, 3, 4	3jca 1128	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)))
6		cdlema2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7		cdlema2.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
8		cdlema2.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
9		cdlema2.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
10	6, 7, 8, 9	exatleN 39449	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄))) → (𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃))
11	5, 10	syld3an3 1411	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (𝑅 ≤ 𝑋 ↔ 𝑅 = 𝑃))
12	11	necon3bbid 2965	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (¬ 𝑅 ≤ 𝑋 ↔ 𝑅 ≠ 𝑃))
13	1, 12	mpbird 257	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → ¬ 𝑅 ≤ 𝑋)
14		simp1l 1198	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝐾 ∈ HL)
15		hlatl 39405	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
16	14, 15	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝐾 ∈ AtLat)
17		simp23 1209	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝑅 ∈ 𝐴)
18		simp1r 1199	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → 𝑋 ∈ 𝐵)
19		cdlema2.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
20		cdlema2.z	. . . 4 ⊢ 0 = (0.‘𝐾)
21	6, 7, 19, 20, 9	atnle 39362	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑅 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑅 ≤ 𝑋 ↔ (𝑅 ∧ 𝑋) = 0 ))
22	16, 17, 18, 21	syl3anc 1373	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (¬ 𝑅 ≤ 𝑋 ↔ (𝑅 ∧ 𝑋) = 0 ))
23	13, 22	mpbid 232	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (𝑅 ∧ 𝑋) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   class class class wbr 5091  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  lecple 17168  joincjn 18217  meetcmee 18218  0.cp0 18327  Atomscatm 39308  AtLatcal 39309  HLchlt 39395

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-proset 18200  df-poset 18219  df-plt 18234  df-lub 18250  df-glb 18251  df-join 18252  df-meet 18253  df-p0 18329  df-lat 18338  df-covers 39311  df-ats 39312  df-atl 39343  df-cvlat 39367  df-hlat 39396

This theorem is referenced by: (None)