Theorem atmod2i2 39844

Description: Version of modular law pmod2iN 39831 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-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
atmod2i2	⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑃) ∨ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑌)))

Proof of Theorem atmod2i2

Step	Hyp	Ref	Expression
1		hllat 39344	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
2	1	3ad2ant1 1132	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ Lat)
3		simp21 1205	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝑃 ∈ 𝐴)
4		atmod.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
5		atmod.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
6	4, 5	atbase 39270	. . . . . 6 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ 𝐵)
7	3, 6	syl 17	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝑃 ∈ 𝐵)
8		simp23 1207	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝑌 ∈ 𝐵)
9		atmod.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
10	4, 9	latjcom 18504	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃 ∨ 𝑌) = (𝑌 ∨ 𝑃))
11	2, 7, 8, 10	syl3anc 1370	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∨ 𝑌) = (𝑌 ∨ 𝑃))
12	11	oveq1d 7445	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑌) ∧ 𝑋) = ((𝑌 ∨ 𝑃) ∧ 𝑋))
13		simp22 1206	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝑋 ∈ 𝐵)
14	4, 9	latjcl 18496	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃 ∨ 𝑌) ∈ 𝐵)
15	2, 7, 8, 14	syl3anc 1370	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∨ 𝑌) ∈ 𝐵)
16		atmod.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
17	4, 16	latmcom 18520	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑌) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑌)) = ((𝑃 ∨ 𝑌) ∧ 𝑋))
18	2, 13, 15, 17	syl3anc 1370	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑌)) = ((𝑃 ∨ 𝑌) ∧ 𝑋))
19		simp1 1135	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ HL)
20		simp3 1137	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → 𝑌 ≤ 𝑋)
21		atmod.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
22	4, 21, 9, 16, 5	atmod1i2 39841	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ (𝑃 ∧ 𝑋)) = ((𝑌 ∨ 𝑃) ∧ 𝑋))
23	19, 3, 8, 13, 20, 22	syl131anc 1382	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ (𝑃 ∧ 𝑋)) = ((𝑌 ∨ 𝑃) ∧ 𝑋))
24	12, 18, 23	3eqtr4d 2784	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑌)) = (𝑌 ∨ (𝑃 ∧ 𝑋)))
25	4, 16	latmcl 18497	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) ∈ 𝐵)
26	2, 7, 13, 25	syl3anc 1370	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∧ 𝑋) ∈ 𝐵)
27	4, 9	latjcom 18504	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ (𝑃 ∧ 𝑋) ∈ 𝐵) → (𝑌 ∨ (𝑃 ∧ 𝑋)) = ((𝑃 ∧ 𝑋) ∨ 𝑌))
28	2, 8, 26, 27	syl3anc 1370	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ (𝑃 ∧ 𝑋)) = ((𝑃 ∧ 𝑋) ∨ 𝑌))
29	4, 16	latmcom 18520	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑋) = (𝑋 ∧ 𝑃))
30	2, 7, 13, 29	syl3anc 1370	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∧ 𝑋) = (𝑋 ∧ 𝑃))
31	30	oveq1d 7445	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∧ 𝑋) ∨ 𝑌) = ((𝑋 ∧ 𝑃) ∨ 𝑌))
32	24, 28, 31	3eqtrrd 2779	1 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑃) ∨ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑌)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   class class class wbr 5147  ‘cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  meetcmee 18369  Latclat 18488  Atomscatm 39244  HLchlt 39331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-proset 18351  df-poset 18370  df-plt 18387  df-lub 18403  df-glb 18404  df-join 18405  df-meet 18406  df-p0 18482  df-lat 18489  df-clat 18556  df-oposet 39157  df-ol 39159  df-oml 39160  df-covers 39247  df-ats 39248  df-atl 39279  df-cvlat 39303  df-hlat 39332  df-psubsp 39485  df-pmap 39486  df-padd 39778

This theorem is referenced by:  llnexchb2lem  39850  dalawlem2  39854  dalawlem3  39855  dalawlem11  39863  dalawlem12  39864  cdleme15b  40257