Theorem llnmod2i2 37877

Description: Version of modular law pmod1i 37862 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 ∨ 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)

Hypotheses

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

Assertion

Ref	Expression
llnmod2i2	⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∨ 𝑌) = (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑌)))

Proof of Theorem llnmod2i2

Step	Hyp	Ref	Expression
1		simp11 1202	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ HL)
2	1	hllatd 37378	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ Lat)
3		simp13 1204	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑌 ∈ 𝐵)
4		simp2l 1198	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑃 ∈ 𝐴)
5		simp2r 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑄 ∈ 𝐴)
6		atmod.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7		atmod.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
8		atmod.a	. . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾)
9	6, 7, 8	hlatjcl 37381	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
10	1, 4, 5, 9	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∨ 𝑄) ∈ 𝐵)
11		simp12 1203	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑋 ∈ 𝐵)
12		atmod.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
13	6, 12	latmcl 18158	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵)
14	2, 10, 11, 13	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵)
15	6, 7	latjcom 18165	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵) → (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)) = (((𝑃 ∨ 𝑄) ∧ 𝑋) ∨ 𝑌))
16	2, 3, 14, 15	syl3anc 1370	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)) = (((𝑃 ∨ 𝑄) ∧ 𝑋) ∨ 𝑌))
17	6, 7	latjcl 18157	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
18	2, 3, 10, 17	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
19	6, 12	latmcom 18181	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄))) = ((𝑌 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑋))
20	2, 11, 18, 19	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄))) = ((𝑌 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑋))
21	6, 7	latjcom 18165	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∨ 𝑌) = (𝑌 ∨ (𝑃 ∨ 𝑄)))
22	2, 10, 3, 21	syl3anc 1370	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑄) ∨ 𝑌) = (𝑌 ∨ (𝑃 ∨ 𝑄)))
23	22	oveq2d 7291	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑌)) = (𝑋 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄))))
24		simp3 1137	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑌 ≤ 𝑋)
25		atmod.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
26	6, 25, 7, 12, 8	llnmod1i2 37874	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)) = ((𝑌 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑋))
27	1, 3, 11, 4, 5, 24, 26	syl321anc 1391	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)) = ((𝑌 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑋))
28	20, 23, 27	3eqtr4d 2788	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑌)) = (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)))
29	6, 12	latmcom 18181	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ 𝑋))
30	2, 11, 10, 29	syl3anc 1370	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ 𝑋))
31	30	oveq1d 7290	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∨ 𝑌) = (((𝑃 ∨ 𝑄) ∧ 𝑋) ∨ 𝑌))
32	16, 28, 31	3eqtr4rd 2789	1 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∨ 𝑌) = (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑌)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   class class class wbr 5074  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  lecple 16969  joincjn 18029  meetcmee 18030  Latclat 18149  Atomscatm 37277  HLchlt 37364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-lat 18150  df-clat 18217  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-psubsp 37517  df-pmap 37518  df-padd 37810

This theorem is referenced by:  dalawlem11  37895