Theorem dihmeetlem7N 41419

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 1192	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ HL)
3		hlatl 39469	. . . . . 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 1194	. . . . 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 2733	. . . . . 6 ⊢ (0.‘𝐾) = (0.‘𝐾)
11		dihmeetlem7.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
12	7, 8, 9, 10, 11	atnle 39426	. . . . 5 ⊢ ((𝐾 ∈ AtLat ∧ 𝑝 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = (0.‘𝐾)))
13	4, 5, 6, 12	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (¬ 𝑝 ≤ 𝑌 ↔ (𝑝 ∧ 𝑌) = (0.‘𝐾)))
14	1, 13	mpbid 232	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑝 ∧ 𝑌) = (0.‘𝐾))
15	14	oveq2d 7371	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)))
16	2	hllatd 39473	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝐾 ∈ Lat)
17		simpl2 1193	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → 𝑋 ∈ 𝐵)
18	7, 9	latmcl 18356	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
19	16, 17, 6, 18	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (𝑋 ∧ 𝑌) ∈ 𝐵)
20	7, 8, 9	latmle2 18381	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
21	16, 17, 6, 20	syl3anc 1373	. . 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 39968	. . 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 1385	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → ((𝑋 ∧ 𝑌) ∨ (𝑝 ∧ 𝑌)) = (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌))
25		hlol 39470	. . . 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 39334	. . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑌) ∈ 𝐵) → ((𝑋 ∧ 𝑌) ∨ (0.‘𝐾)) = (𝑋 ∧ 𝑌))
28	26, 19, 27	syl2anc 584	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5095  ‘cfv 6489  (class class class)co 7355  Basecbs 17130  lecple 17178  joincjn 18227  meetcmee 18228  0.cp0 18337  Latclat 18347  OLcol 39283  Atomscatm 39372  AtLatcal 39373  HLchlt 39459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-iin 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-proset 18210  df-poset 18229  df-plt 18244  df-lub 18260  df-glb 18261  df-join 18262  df-meet 18263  df-p0 18339  df-lat 18348  df-clat 18415  df-oposet 39285  df-ol 39287  df-oml 39288  df-covers 39375  df-ats 39376  df-atl 39407  df-cvlat 39431  df-hlat 39460  df-psubsp 39612  df-pmap 39613  df-padd 39905

This theorem is referenced by: (None)