Theorem atnle 39335

Description: Two ways of expressing "an atom is not less than or equal to a lattice element." (atnssm0 32357 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 1192	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → 𝐾 ∈ AtLat)
2		atllat 39318	. . . . . . . . 9 ⊢ (𝐾 ∈ AtLat → 𝐾 ∈ Lat)
3	2	3ad2ant1 1133	. . . . . . . 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 39307	. . . . . . . . 9 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	6	3ad2ant2 1134	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑃 ∈ 𝐵)
8		simp3 1138	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵)
9		atnle.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
10	4, 9	latmcl 18450	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
11	3, 7, 8, 10	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
12	11	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → (𝑃 ∧ 𝑋) ∈ 𝐵)
13		simpr 484	. . . . . 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 39334	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ (𝑃 ∧ 𝑋) ∈ 𝐵 ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋))
17	1, 12, 13, 16	syl3anc 1373	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) → ∃𝑦 ∈ 𝐴 𝑦 ≤ (𝑃 ∧ 𝑋))
18		simpl1 1192	. . . . . . . . . 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 39307	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐵)
21	20	adantl 481	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐵)
22		simpl2 1193	. . . . . . . . . 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 1194	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑋 ∈ 𝐵)
25	4, 14, 9	latlem12 18476	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝑦 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) ↔ 𝑦 ≤ (𝑃 ∧ 𝑋)))
26	19, 21, 23, 24, 25	syl13anc 1374	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) ↔ 𝑦 ≤ (𝑃 ∧ 𝑋)))
27		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
28	14, 5	atcmp 39329	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ AtLat ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑦 ≤ 𝑃 ↔ 𝑦 = 𝑃))
29	18, 27, 22, 28	syl3anc 1373	. . . . . . . . . 10 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑃 ↔ 𝑦 = 𝑃))
30		breq1 5122	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑃 → (𝑦 ≤ 𝑋 ↔ 𝑃 ≤ 𝑋))
31	30	biimpd 229	. . . . . . . . . 10 ⊢ (𝑦 = 𝑃 → (𝑦 ≤ 𝑋 → 𝑃 ≤ 𝑋))
32	29, 31	biimtrdi 253	. . . . . . . . 9 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ 𝑃 → (𝑦 ≤ 𝑋 → 𝑃 ≤ 𝑋)))
33	32	impd 410	. . . . . . . 8 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → ((𝑦 ≤ 𝑃 ∧ 𝑦 ≤ 𝑋) → 𝑃 ≤ 𝑋))
34	26, 33	sylbird 260	. . . . . . 7 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
35	34	adantlr 715	. . . . . 6 ⊢ ((((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) ≠ 0 ) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤ (𝑃 ∧ 𝑋) → 𝑃 ≤ 𝑋))
36	35	rexlimdva 3141	. . . . 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 412	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑋) ≠ 0 → 𝑃 ≤ 𝑋))
39	38	necon1bd 2950	. 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 → (𝑃 ∧ 𝑋) = 0 ))
40	15, 5	atn0 39326	. . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → 𝑃 ≠ 0 )
41	40	3adant3 1132	. . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → 𝑃 ≠ 0 )
42	4, 14, 9	latleeqm1 18477	. . . . . . . 8 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
43	3, 7, 8, 42	syl3anc 1373	. . . . . . 7 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
44	43	adantr 480	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 𝑃))
45		eqeq1 2739	. . . . . . . 8 ⊢ ((𝑃 ∧ 𝑋) = 𝑃 → ((𝑃 ∧ 𝑋) = 0 ↔ 𝑃 = 0 ))
46	45	biimpcd 249	. . . . . . 7 ⊢ ((𝑃 ∧ 𝑋) = 0 → ((𝑃 ∧ 𝑋) = 𝑃 → 𝑃 = 0 ))
47	46	adantl 481	. . . . . 6 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → ((𝑃 ∧ 𝑋) = 𝑃 → 𝑃 = 0 ))
48	44, 47	sylbid 240	. . . . 5 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≤ 𝑋 → 𝑃 = 0 ))
49	48	necon3ad 2945	. . . 4 ⊢ (((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∧ 𝑋) = 0 ) → (𝑃 ≠ 0 → ¬ 𝑃 ≤ 𝑋))
50	49	ex 412	. . 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 212	1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (¬ 𝑃 ≤ 𝑋 ↔ (𝑃 ∧ 𝑋) = 0 ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∃wrex 3060   class class class wbr 5119  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  lecple 17278  meetcmee 18324  0.cp0 18433  Latclat 18441  Atomscatm 39281  AtLatcal 39282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-proset 18306  df-poset 18325  df-plt 18340  df-lub 18356  df-glb 18357  df-join 18358  df-meet 18359  df-p0 18435  df-lat 18442  df-covers 39284  df-ats 39285  df-atl 39316

This theorem is referenced by:  atnem0  39336  iscvlat2N  39342  cvlexch3  39350  cvlexch4N  39351  cvlcvrp  39358  intnatN  39426  cvrat4  39462  dalem24  39716  cdlema2N  39811  llnexchb2lem  39887  lhpmat  40049  cdleme15b  40294  cdlemednpq  40318  cdleme20zN  40320  cdleme22cN  40361  dihmeetlem7N  41329  dihmeetlem17N  41342