Theorem atnle 40190

Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 32857 analog.) (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
atnle.b	⊢ 𝐵 = (Base‘𝐾)
atnle.l	⊢ ≤ = (le‘𝐾)
atnle.m	⊢ ∧ = (meet‘𝐾)
atnle.z	⊢ 0 = (0.‘𝐾)
atnle.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
atnle	⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))

Proof of Theorem atnle

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1210	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2		atllat 40173	. . . . . . . . 9 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
3	2	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat)
4		atnle.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐾)
5		atnle.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 40162	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	6	3ad2ant2 1152	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑃 ∈ 𝐵)
8		simp3 1156	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
9		atnle.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
10	4, 9	latmcl 18528	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
11	3, 7, 8, 10	syl3anc 1398	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
12	11	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → (𝑃 ∧ 𝑋) ∈ 𝐵)
13		simpr 490	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → (𝑃 ∧ 𝑋) ≠ 0 )
14		atnle.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
15		atnle.z	. . . . . . 7 ⊢ 0 = (0.‘𝐾)
16	4, 14, 15, 5	atlex 40189	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∧ 𝑋) ∈ 𝐵 ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋))
17	1, 12, 13, 16	syl3anc 1398	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋))
18		simpl1 1210	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝐾 ∈ AtLat)
19	18, 2	syl 18	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝐾 ∈ Lat)
20	4, 5	atbase 40162	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵)
21	20	adantl 487	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
22		simpl2 1211	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑃 ∈ 𝐴)
23	22, 6	syl 18	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑃 ∈ 𝐵)
24		simpl3 1212	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑋 ∈ 𝐵)
25	4, 14, 9	latlem12 18554	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑦 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) ↔ 𝑦 ≤ (𝑃 ∧ 𝑋)))
26	19, 21, 23, 24, 25	syl13anc 1399	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) ↔ 𝑦 ≤ (𝑃 ∧ 𝑋)))
27		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
28	14, 5	atcmp 40184	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑦 ≤ 𝑃 ↔ 𝑦 = 𝑃))
29	18, 27, 22, 28	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑃 ↔ 𝑦 = 𝑃))
30		breq1 5106	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑃 → (𝑦 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
31	30	biimpd 232	. . . . . . . . . 10 ⊢ (𝑦 = 𝑃 → (𝑦 ≤ 𝑋 → 𝑃 ≤ 𝑋))
32	29, 31	biimtrdi 256	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑃 → (𝑦 ≤ 𝑋 → 𝑃 ≤ 𝑋)))
33	32	impd 416	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) → 𝑃 ≤ 𝑋))
34	26, 33	sylbird 263	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
35	34	adantlr 728	. . . . . 6 ⊢ ((((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
36	35	rexlimdva 3163	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → (∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
37	17, 36	mpd 16	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → 𝑃 ≤ 𝑋)
38	37	ex 418	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑋) ≠ 0 → 𝑃 ≤ 𝑋))
39	38	necon1bd 2973	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ∧ 𝑋) = 0 ))
40	15, 5	atn0 40181	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 )
41	40	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑃 ≠ 0 )
42	4, 14, 9	latleeqm1 18555	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
43	3, 7, 8, 42	syl3anc 1398	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
44	43	adantr 486	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
45		eqeq1 2764	. . . . . . . 8 ⊢ ((𝑃 ∧ 𝑋) = 𝑃 → ((𝑃 ∧ 𝑋) = 0 ↔ 𝑃 = 0 ))
46	45	biimpcd 252	. . . . . . 7 ⊢ ((𝑃 ∧ 𝑋) = 0 → ((𝑃 ∧ 𝑋) = 𝑃 → 𝑃 = 0 ))
47	46	adantl 487	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → ((𝑃 ∧ 𝑋) = 𝑃 → 𝑃 = 0 ))
48	44, 47	sylbid 243	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ 𝑋 → 𝑃 = 0 ))
49	48	necon3ad 2968	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≠ 0 → ¬ 𝑃 ≤ 𝑋))
50	49	ex 418	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑋) = 0 → (𝑃 ≠ 0 → ¬ 𝑃 ≤ 𝑋)))
51	41, 50	mpid 45	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑋) = 0 → ¬ 𝑃 ≤ 𝑋))
52	39, 51	impbid 215	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∃wrex 3086   class class class wbr 5103  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  lecple 17349  meetcmee 18400  0.cp0 18509  Latclat 18519  Atomscatm 40136  AtLatcal 40137

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-proset 18382  df-poset 18401  df-plt 18416  df-lub 18432  df-glb 18433  df-join 18434  df-meet 18435  df-p0 18511  df-lat 18520  df-covers 40139  df-ats 40140  df-atl 40171

This theorem is used by:  atnem0  40191  iscvlat2N  40197  cvlexch3  40205  cvlexch4N  40206  cvlcvrp  40213  intnatN  40280  cvrat4  40316  dalem24  40570  cdlema2N  40665  llnexchb2lem  40741  lhpmat  40903  cdleme15b  41148  cdlemednpq  41172  cdleme20zN  41174  cdleme22cN  41215  dihmeetlem7N  42183  dihmeetlem17N  42196