Theorem atnle 40374

Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 32978 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 40357	. . . . . . . . 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 40346	. . . . . . . . 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 18614	. . . . . . . 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 40373	. . . . . 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 40346	. . . . . . . . . 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 18640	. . . . . . . . 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 40368	. . . . . . . . . . 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 3164	. . . . 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 2974	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ∧ 𝑋) = 0 ))
40	15, 5	atn0 40365	. . . 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 18641	. . . . . . . 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 2765	. . . . . . . 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 2969	. . . 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 2956  ∃wrex 3087   class class class wbr 5103  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  lecple 17435  meetcmee 18486  0.cp0 18595  Latclat 18605  Atomscatm 40320  AtLatcal 40321

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-lat 18606  df-covers 40323  df-ats 40324  df-atl 40355

This theorem is used by:  atnem0  40375  iscvlat2N  40381  cvlexch3  40389  cvlexch4N  40390  cvlcvrp  40397  intnatN  40464  cvrat4  40500  dalem24  40754  cdlema2N  40849  llnexchb2lem  40925  lhpmat  41087  cdleme15b  41332  cdlemednpq  41356  cdleme20zN  41358  cdleme22cN  41399  dihmeetlem7N  42367  dihmeetlem17N  42380