Theorem cvrat 37363

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 30649 analog.) (Contributed by NM, 22-Nov-2011.)

Hypotheses

Ref	Expression
cvrat.b	⊢ 𝐵 = (Base‘𝐾)
cvrat.s	⊢ < = (lt‘𝐾)
cvrat.j	⊢ ∨ = (join‘𝐾)
cvrat.z	⊢ 0 = (0.‘𝐾)
cvrat.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
cvrat	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝐴))

Proof of Theorem cvrat

Step	Hyp	Ref	Expression
1		cvrat.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		cvrat.s	. . . 4 ⊢ < = (lt‘𝐾)
3		cvrat.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		cvrat.z	. . . 4 ⊢ 0 = (0.‘𝐾)
5		cvrat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6	1, 2, 3, 4, 5	cvratlem 37362	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴))
7		hllat 37304	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
9		simpr2 1193	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
10	1, 5	atbase 37230	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
12		simpr3 1194	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
13	1, 5	atbase 37230	. . . . . . . . 9 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
14	12, 13	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
15	1, 3	latjcom 18080	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
16	8, 11, 14, 15	syl3anc 1369	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
17	16	breq2d 5082	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 < (𝑃 ∨ 𝑄) ↔ 𝑋 < (𝑄 ∨ 𝑃)))
18	17	anbi2d 628	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) ↔ (𝑋 ≠ 0 ∧ 𝑋 < (𝑄 ∨ 𝑃))))
19		simpl 482	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ HL)
20		simpr1 1192	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
21	1, 2, 3, 4, 5	cvratlem 37362	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑄 ∨ 𝑃))) → (¬ 𝑄(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴))
22	21	ex 412	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑄 ∨ 𝑃)) → (¬ 𝑄(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴)))
23	19, 20, 12, 9, 22	syl13anc 1370	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑄 ∨ 𝑃)) → (¬ 𝑄(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴)))
24	18, 23	sylbid 239	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) → (¬ 𝑄(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴)))
25	24	imp 406	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) → (¬ 𝑄(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴))
26		hlpos 37307	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Poset)
28	1, 3	latjcl 18072	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
29	8, 11, 14, 28	syl3anc 1369	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
30		eqid 2738	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
31	1, 30, 2	pltnle 17971	. . . . . . . . 9 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) ∧ 𝑋 < (𝑃 ∨ 𝑄)) → ¬ (𝑃 ∨ 𝑄)(le‘𝐾)𝑋)
32	31	ex 412	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 < (𝑃 ∨ 𝑄) → ¬ (𝑃 ∨ 𝑄)(le‘𝐾)𝑋))
33	27, 20, 29, 32	syl3anc 1369	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 < (𝑃 ∨ 𝑄) → ¬ (𝑃 ∨ 𝑄)(le‘𝐾)𝑋))
34	1, 30, 3	latjle12 18083	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑃(le‘𝐾)𝑋 ∧ 𝑄(le‘𝐾)𝑋) ↔ (𝑃 ∨ 𝑄)(le‘𝐾)𝑋))
35	8, 11, 14, 20, 34	syl13anc 1370	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃(le‘𝐾)𝑋 ∧ 𝑄(le‘𝐾)𝑋) ↔ (𝑃 ∨ 𝑄)(le‘𝐾)𝑋))
36	35	biimpd 228	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑃(le‘𝐾)𝑋 ∧ 𝑄(le‘𝐾)𝑋) → (𝑃 ∨ 𝑄)(le‘𝐾)𝑋))
37	33, 36	nsyld 156	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 < (𝑃 ∨ 𝑄) → ¬ (𝑃(le‘𝐾)𝑋 ∧ 𝑄(le‘𝐾)𝑋)))
38		ianor 978	. . . . . 6 ⊢ (¬ (𝑃(le‘𝐾)𝑋 ∧ 𝑄(le‘𝐾)𝑋) ↔ (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
39	37, 38	syl6ib 250	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 < (𝑃 ∨ 𝑄) → (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋)))
40	39	imp 406	. . . 4 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ 𝑋 < (𝑃 ∨ 𝑄)) → (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
41	40	adantrl 712	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
42	6, 25, 41	mpjaod 856	. 2 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) → 𝑋 ∈ 𝐴)
43	42	ex 412	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄)) → 𝑋 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  lecple 16895  Posetcpo 17940  ltcplt 17941  joincjn 17944  0.cp0 18056  Latclat 18064  Atomscatm 37204  HLchlt 37291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-lat 18065  df-clat 18132  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292

This theorem is referenced by:  cvrat2  37370