Theorem cvrat 39441

Description: A nonzero Hilbert lattice element less than the join of two atoms is an atom. (atcvati 32367 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 39440	. . 3 ⊢ (((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) ∧ (𝑋 ≠ 0 ∧ 𝑋 < (𝑃 ∨ 𝑄))) → (¬ 𝑃(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴))
7		hllat 39381	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
8	7	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Lat)
9		simpr2 1196	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐴)
10	1, 5	atbase 39307	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑃 ∈ 𝐵)
12		simpr3 1197	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
13	1, 5	atbase 39307	. . . . . . . . 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 18457	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
16	8, 11, 14, 15	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) = (𝑄 ∨ 𝑃))
17	16	breq2d 5131	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 < (𝑃 ∨ 𝑄) ↔ 𝑋 < (𝑄 ∨ 𝑃)))
18	17	anbi2d 630	. . . . 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 1195	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
21	1, 2, 3, 4, 5	cvratlem 39440	. . . . . . 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 1374	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → ((𝑋 ≠ 0 ∧ 𝑋 < (𝑄 ∨ 𝑃)) → (¬ 𝑄(le‘𝐾)𝑋 → 𝑋 ∈ 𝐴)))
24	18, 23	sylbid 240	. . . 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 39384	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Poset)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → 𝐾 ∈ Poset)
28	1, 3	latjcl 18449	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 ∨ 𝑄) ∈ 𝐵)
29	8, 11, 14, 28	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑃 ∨ 𝑄) ∈ 𝐵)
30		eqid 2735	. . . . . . . . . 10 ⊢ (le‘𝐾) = (le‘𝐾)
31	1, 30, 2	pltnle 18348	. . . . . . . . 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 1373	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) → (𝑋 < (𝑃 ∨ 𝑄) → ¬ (𝑃 ∨ 𝑄)(le‘𝐾)𝑋))
34	1, 30, 3	latjle12 18460	. . . . . . . . 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 1374	. . . . . . . 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 229	. . . . . . 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 983	. . . . . 6 ⊢ (¬ (𝑃(le‘𝐾)𝑋 ∧ 𝑄(le‘𝐾)𝑋) ↔ (¬ 𝑃(le‘𝐾)𝑋 ∨ ¬ 𝑄(le‘𝐾)𝑋))
39	37, 38	imbitrdi 251	. . . . 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 716	. . 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 860	. 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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  lecple 17278  Posetcpo 18319  ltcplt 18320  joincjn 18323  0.cp0 18433  Latclat 18441  Atomscatm 39281  HLchlt 39368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-lat 18442  df-clat 18509  df-oposet 39194  df-ol 39196  df-oml 39197  df-covers 39284  df-ats 39285  df-atl 39316  df-cvlat 39340  df-hlat 39369

This theorem is referenced by:  cvrat2  39448