Theorem intnatN 39899

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 39852	. . . . . . 7 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
2	1	3ad2ant1 1139	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ AtLat)
3	2	ad2antrr 732	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → 𝐾 ∈ AtLat)
4		eqid 2739	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
5		intnat.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atn0 39800	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → (𝑋 ∧ 𝑌) ≠ (0.‘𝐾))
7	3, 6	sylancom 594	. . . 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 1219	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ HL)
10	9	hllatd 39856	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝐾 ∈ Lat)
11		simpll2 1220	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → 𝑋 ∈ 𝐵)
12		simpll3 1221	. . . . . . 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 18420	. . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))
16	10, 11, 12, 15	syl3anc 1379	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))
17		simplr 774	. . . . . . 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 39809	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 ↔ (𝑌 ∧ 𝑋) = (0.‘𝐾)))
22	18, 19, 11, 21	syl3anc 1379	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (¬ 𝑌 ≤ 𝑋 ↔ (𝑌 ∧ 𝑋) = (0.‘𝐾)))
23	17, 22	mpbid 233	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ 𝑌 ≤ 𝑋) ∧ 𝑌 ∈ 𝐴) → (𝑌 ∧ 𝑋) = (0.‘𝐾))
24	16, 23	eqtrd 2774	. . . . 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 2947	. . 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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  Basecbs 17170  lecple 17218  meetcmee 18269  0.cp0 18378  Latclat 18388  Atomscatm 39755  AtLatcal 39756  HLchlt 39842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-proset 18251  df-poset 18270  df-plt 18285  df-lub 18301  df-glb 18302  df-join 18303  df-meet 18304  df-p0 18380  df-lat 18389  df-covers 39758  df-ats 39759  df-atl 39790  df-cvlat 39814  df-hlat 39843

This theorem is referenced by: (None)