Theorem atnle 36613

Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 30159 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 1188	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2		atllat 36596	. . . . . . . . 9 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
3	2	3ad2ant1 1130	. . . . . . . 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 36585	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	6	3ad2ant2 1131	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑃 ∈ 𝐵)
8		simp3 1135	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
9		atnle.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
10	4, 9	latmcl 17654	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
11	3, 7, 8, 10	syl3anc 1368	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
12	11	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → (𝑃 ∧ 𝑋) ∈ 𝐵)
13		simpr 488	. . . . . 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 36612	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∧ 𝑋) ∈ 𝐵 ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋))
17	1, 12, 13, 16	syl3anc 1368	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋))
18		simpl1 1188	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝐾 ∈ AtLat)
19	18, 2	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝐾 ∈ Lat)
20	4, 5	atbase 36585	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵)
21	20	adantl 485	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
22		simpl2 1189	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑃 ∈ 𝐴)
23	22, 6	syl 17	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑃 ∈ 𝐵)
24		simpl3 1190	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑋 ∈ 𝐵)
25	4, 14, 9	latlem12 17680	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑦 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) ↔ 𝑦 ≤ (𝑃 ∧ 𝑋)))
26	19, 21, 23, 24, 25	syl13anc 1369	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) ↔ 𝑦 ≤ (𝑃 ∧ 𝑋)))
27		simpr 488	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
28	14, 5	atcmp 36607	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑦 ≤ 𝑃 ↔ 𝑦 = 𝑃))
29	18, 27, 22, 28	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑃 ↔ 𝑦 = 𝑃))
30		breq1 5033	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑃 → (𝑦 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
31	30	biimpd 232	. . . . . . . . . 10 ⊢ (𝑦 = 𝑃 → (𝑦 ≤ 𝑋 → 𝑃 ≤ 𝑋))
32	29, 31	syl6bi 256	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑃 → (𝑦 ≤ 𝑋 → 𝑃 ≤ 𝑋)))
33	32	impd 414	. . . . . . . 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 714	. . . . . 6 ⊢ ((((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
36	35	rexlimdva 3243	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → (∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
37	17, 36	mpd 15	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → 𝑃 ≤ 𝑋)
38	37	ex 416	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑋) ≠ 0 → 𝑃 ≤ 𝑋))
39	38	necon1bd 3005	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ∧ 𝑋) = 0 ))
40	15, 5	atn0 36604	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 )
41	40	3adant3 1129	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑃 ≠ 0 )
42	4, 14, 9	latleeqm1 17681	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
43	3, 7, 8, 42	syl3anc 1368	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
44	43	adantr 484	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
45		eqeq1 2802	. . . . . . . 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 485	. . . . . 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 3000	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≠ 0 → ¬ 𝑃 ≤ 𝑋))
50	49	ex 416	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑋) = 0 → (𝑃 ≠ 0 → ¬ 𝑃 ≤ 𝑋)))
51	41, 50	mpid 44	. 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 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∃wrex 3107   class class class wbr 5030  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  lecple 16564  meetcmee 17547  0.cp0 17639  Latclat 17647  Atomscatm 36559  AtLatcal 36560

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-covers 36562  df-ats 36563  df-atl 36594

This theorem is referenced by:  atnem0  36614  iscvlat2N  36620  cvlexch3  36628  cvlexch4N  36629  cvlcvrp  36636  intnatN  36703  cvrat4  36739  dalem24  36993  cdlema2N  37088  llnexchb2lem  37164  lhpmat  37326  cdleme15b  37571  cdlemednpq  37595  cdleme20zN  37597  cdleme22cN  37638  dihmeetlem7N  38606  dihmeetlem17N  38619