Theorem intnatN 37943

Description: If the intersection with a non-majorizing element is an atom, the intersecting element is not an atom. (Contributed by NM, 26-Jun-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
intnat.b	⊢ 𝐵 = (Base‘𝐾)
intnat.l	⊢ ≤ = (le‘𝐾)
intnat.m	⊢ ∧ = (meet‘𝐾)
intnat.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
intnatN	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑌 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ∈ 𝐴)) → ¬ 𝑌 ∈ 𝐴)

Proof of Theorem intnatN

Step	Hyp	Ref	Expression
1		hlatl 37895	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	3ad2ant1 1133	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ AtLat)
3	2	ad2antrr 724	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4		eqid 2731	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		intnat.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atn0 37843	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾))
7	3, 6	sylancom 588	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾))
8	7	ex 413	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∈ 𝐴 → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾)))
9		simpll1 1212	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ HL)
10	9	hllatd 37899	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ Lat)
11		simpll2 1213	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐵)
12		simpll3 1214	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐵)
13		intnat.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
14		intnat.m	. . . . . . . 8 ⊢ ∧ = (meet‘𝐾)
15	13, 14	latmcom 18366	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))
16	10, 11, 12, 15	syl3anc 1371	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))
17		simplr 767	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → ¬ 𝑌 ≤ 𝑋)
18	9, 1	syl 17	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ AtLat)
19		simpr 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑌 ∈ 𝐴)
20		intnat.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
21	13, 20, 14, 4, 5	atnle 37852	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 ↔ (𝑌 ∧ 𝑋) = (0.‘𝐾)))
22	18, 19, 11, 21	syl3anc 1371	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (¬ 𝑌 ≤ 𝑋 ↔ (𝑌 ∧ 𝑋) = (0.‘𝐾)))
23	17, 22	mpbid 231	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∧ 𝑋) = (0.‘𝐾))
24	16, 23	eqtrd 2771	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑋 ∧ 𝑌) = (0.‘𝐾))
25	24	ex 413	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → (𝑌 ∈ 𝐴 → (𝑋 ∧ 𝑌) = (0.‘𝐾)))
26	25	necon3ad 2952	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ≠ (0.‘𝐾) → ¬ 𝑌 ∈ 𝐴))
27	8, 26	syld 47	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∈ 𝐴 → ¬ 𝑌 ∈ 𝐴))
28	27	impr 455	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (¬ 𝑌 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ∈ 𝐴)) → ¬ 𝑌 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ 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  meetcmee 18215  0.cp0 18326  Latclat 18334  Atomscatm 37798  AtLatcal 37799  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-lat 18335  df-covers 37801  df-ats 37802  df-atl 37833  df-cvlat 37857  df-hlat 37886

This theorem is referenced by: (None)