Theorem dihmeetlem7N 40783

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dihmeetlem7.b	⊢ 𝐵 = (Base‘𝐾)
dihmeetlem7.l	⊢ ≤ = (le‘𝐾)
dihmeetlem7.j	⊢ ∨ = (join‘𝐾)
dihmeetlem7.m	⊢ ∧ = (meet‘𝐾)
dihmeetlem7.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
dihmeetlem7N	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌) = (𝑋 ∧ 𝑌))

Proof of Theorem dihmeetlem7N

Step	Hyp	Ref	Expression
1		simprr 772	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ¬ 𝑝 ≤ 𝑌)
2		simpl1 1189	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ HL)
3		hlatl 38832	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4	2, 3	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ AtLat)
5		simprl 770	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝑝 ∈ 𝐴)
6		simpl3 1191	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝑌 ∈ 𝐵)
7		dihmeetlem7.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
8		dihmeetlem7.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
9		dihmeetlem7.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
10		eqid 2728	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
11		dihmeetlem7.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
12	7, 8, 9, 10, 11	atnle 38789	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = (0.‘𝐾)))
13	4, 5, 6, 12	syl3anc 1369	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = (0.‘𝐾)))
14	1, 13	mpbid 231	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑝 ∧ 𝑌) = (0.‘𝐾))
15	14	oveq2d 7436	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)))
16	2	hllatd 38836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ Lat)
17		simpl2 1190	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
18	7, 9	latmcl 18432	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
19	16, 17, 6, 18	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
20	7, 8, 9	latmle2 18457	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
21	16, 17, 6, 20	syl3anc 1369	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ≤ 𝑌)
22		dihmeetlem7.j	. . . 4 ⊢ ∨ = (join‘𝐾)
23	7, 8, 22, 9, 11	atmod1i2 39332	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑝 ∈ 𝐴 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ 𝑌) ≤ 𝑌) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌))
24	2, 5, 19, 6, 21, 23	syl131anc 1381	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌))
25		hlol 38833	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
26	2, 25	syl 17	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ OL)
27	7, 22, 10	olj01 38697	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)) = (𝑋 ∧ 𝑌))
28	26, 19, 27	syl2anc 583	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)) = (𝑋 ∧ 𝑌))
29	15, 24, 28	3eqtr3d 2776	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌) = (𝑋 ∧ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   class class class wbr 5148  ‘cfv 6548  (class class class)co 7420  Basecbs 17180  lecple 17240  joincjn 18303  meetcmee 18304  0.cp0 18415  Latclat 18423  OLcol 38646  Atomscatm 38735  AtLatcal 38736  HLchlt 38822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-proset 18287  df-poset 18305  df-plt 18322  df-lub 18338  df-glb 18339  df-join 18340  df-meet 18341  df-p0 18417  df-lat 18424  df-clat 18491  df-oposet 38648  df-ol 38650  df-oml 38651  df-covers 38738  df-ats 38739  df-atl 38770  df-cvlat 38794  df-hlat 38823  df-psubsp 38976  df-pmap 38977  df-padd 39269

This theorem is referenced by: (None)