Theorem 1cvrat 38012

Description: Create an atom under an element covered by the lattice unity. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)

Hypotheses

Ref	Expression
1cvrat.b	⊢ 𝐵 = (Base‘𝐾)
1cvrat.l	⊢ ≤ = (le‘𝐾)
1cvrat.j	⊢ ∨ = (join‘𝐾)
1cvrat.m	⊢ ∧ = (meet‘𝐾)
1cvrat.u	⊢ 1 = (1.‘𝐾)
1cvrat.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
1cvrat.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
1cvrat	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)

Proof of Theorem 1cvrat

Step	Hyp	Ref	Expression
1		hllat 37898	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	3ad2ant1 1133	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝐾 ∈ Lat)
3		simp21 1206	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑃 ∈ 𝐴)
4		1cvrat.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
5		1cvrat.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 37824	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	3, 6	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑃 ∈ 𝐵)
8		simp22 1207	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑄 ∈ 𝐴)
9	4, 5	atbase 37824	. . . . . 6 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
10	8, 9	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑄 ∈ 𝐵)
11		1cvrat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
12	4, 11	latjcom 18350	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
13	2, 7, 10, 12	syl3anc 1371	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
14	13	oveq1d 7377	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = ((𝑄 ∨ 𝑃) ∧ 𝑋))
15	4, 11	latjcl 18342	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) → (𝑄 ∨ 𝑃) ∈ 𝐵)
16	2, 10, 7, 15	syl3anc 1371	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑄 ∨ 𝑃) ∈ 𝐵)
17		simp23 1208	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑋 ∈ 𝐵)
18		1cvrat.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
19	4, 18	latmcom 18366	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∨ 𝑃) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑄 ∨ 𝑃) ∧ 𝑋) = (𝑋 ∧ (𝑄 ∨ 𝑃)))
20	2, 16, 17, 19	syl3anc 1371	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑄 ∨ 𝑃) ∧ 𝑋) = (𝑋 ∧ (𝑄 ∨ 𝑃)))
21	14, 20	eqtrd 2771	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) = (𝑋 ∧ (𝑄 ∨ 𝑃)))
22		simp1 1136	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝐾 ∈ HL)
23	17, 8, 3	3jca 1128	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴))
24		simp31 1209	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑃 ≠ 𝑄)
25	24	necomd 2995	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑄 ≠ 𝑃)
26		simp33 1211	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ¬ 𝑃 ≤ 𝑋)
27		hlop 37897	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
28	27	3ad2ant1 1133	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝐾 ∈ OP)
29		1cvrat.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
30		1cvrat.u	. . . . . 6 ⊢ 1 = (1.‘𝐾)
31	4, 29, 30	ople1 37726	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ 𝐵) → 𝑄 ≤ 1 )
32	28, 10, 31	syl2anc 584	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑄 ≤ 1 )
33		simp32 1210	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑋𝐶 1 )
34		1cvrat.c	. . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾)
35	4, 29, 11, 30, 34, 5	1cvrjat 38011	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑋 ∨ 𝑃) = 1 )
36	22, 17, 3, 33, 26, 35	syl32anc 1378	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑋 ∨ 𝑃) = 1 )
37	32, 36	breqtrrd 5138	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → 𝑄 ≤ (𝑋 ∨ 𝑃))
38	4, 29, 11, 18, 5	cvrat3 37978	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) → ((𝑄 ≠ 𝑃 ∧ ¬ 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ (𝑋 ∨ 𝑃)) → (𝑋 ∧ (𝑄 ∨ 𝑃)) ∈ 𝐴))
39	38	imp 407	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) ∧ (𝑄 ≠ 𝑃 ∧ ¬ 𝑃 ≤ 𝑋 ∧ 𝑄 ≤ (𝑋 ∨ 𝑃))) → (𝑋 ∧ (𝑄 ∨ 𝑃)) ∈ 𝐴)
40	22, 23, 25, 26, 37, 39	syl23anc 1377	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → (𝑋 ∧ (𝑄 ∨ 𝑃)) ∈ 𝐴)
41	21, 40	eqeltrd 2832	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ≠ 𝑄 ∧ 𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋)) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2939   class class class wbr 5110  ‘cfv 6501  (class class class)co 7362  Basecbs 17094  lecple 17154  joincjn 18214  meetcmee 18215  1.cp1 18327  Latclat 18334  OPcops 37707   ⋖ ccvr 37797  Atomscatm 37798  HLchlt 37885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-proset 18198  df-poset 18216  df-plt 18233  df-lub 18249  df-glb 18250  df-join 18251  df-meet 18252  df-p0 18328  df-p1 18329  df-lat 18335  df-clat 18402  df-oposet 37711  df-ol 37713  df-oml 37714  df-covers 37801  df-ats 37802  df-atl 37833  df-cvlat 37857  df-hlat 37886

This theorem is referenced by:  cdlemblem  38329  cdlemb  38330  lhpat  38579