Theorem dihmeetlem7N 42185

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 785	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ¬ 𝑝 ≤ 𝑌)
2		simpl1 1210	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ HL)
3		hlatl 40235	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
4	2, 3	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ AtLat)
5		simprl 783	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝑝 ∈ 𝐴)
6		simpl3 1212	. . . . 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 2762	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
11		dihmeetlem7.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
12	7, 8, 9, 10, 11	atnle 40192	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = (0.‘𝐾)))
13	4, 5, 6, 12	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = (0.‘𝐾)))
14	1, 13	mpbid 235	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑝 ∧ 𝑌) = (0.‘𝐾))
15	14	oveq2d 7432	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)))
16	2	hllatd 40239	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ Lat)
17		simpl2 1211	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
18	7, 9	latmcl 18532	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
19	16, 17, 6, 18	syl3anc 1398	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
20	7, 8, 9	latmle2 18557	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
21	16, 17, 6, 20	syl3anc 1398	. . 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 40734	. . 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 1410	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌))
25		hlol 40236	. . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL)
26	2, 25	syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ OL)
27	7, 22, 10	olj01 40100	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)) = (𝑋 ∧ 𝑌))
28	26, 19, 27	syl2anc 596	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)) = (𝑋 ∧ 𝑌))
29	15, 24, 28	3eqtr3d 2805	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌) = (𝑋 ∧ 𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   class class class wbr 5107  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  lecple 17353  joincjn 18403  meetcmee 18404  0.cp0 18513  Latclat 18523  OLcol 40049  Atomscatm 40138  AtLatcal 40139  HLchlt 40225

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-lat 18524  df-clat 18591  df-oposet 40051  df-ol 40053  df-oml 40054  df-covers 40141  df-ats 40142  df-atl 40173  df-cvlat 40197  df-hlat 40226  df-psubsp 40378  df-pmap 40379  df-padd 40671

This theorem is used by: (None)