Theorem llnmod2i2 38326

Description: Version of modular law pmod1i 38311 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 1203	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ HL)
2	1	hllatd 37826	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ Lat)
3		simp13 1205	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑌 ∈ 𝐵)
4		simp2l 1199	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑃 ∈ 𝐴)
5		simp2r 1200	. . . . 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 37829	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
10	1, 4, 5, 9	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∨ 𝑄) ∈ 𝐵)
11		simp12 1204	. . . 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 18329	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵)
14	2, 10, 11, 13	syl3anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵)
15	6, 7	latjcom 18336	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵) → (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)) = (((𝑃 ∨ 𝑄) ∧ 𝑋) ∨ 𝑌))
16	2, 3, 14, 15	syl3anc 1371	. 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 18328	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
18	2, 3, 10, 17	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
19	6, 12	latmcom 18352	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄))) = ((𝑌 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑋))
20	2, 11, 18, 19	syl3anc 1371	. . 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 18336	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∨ 𝑌) = (𝑌 ∨ (𝑃 ∨ 𝑄)))
22	2, 10, 3, 21	syl3anc 1371	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑄) ∨ 𝑌) = (𝑌 ∨ (𝑃 ∨ 𝑄)))
23	22	oveq2d 7373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑌)) = (𝑋 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄))))
24		simp3 1138	. . . 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 38323	. . . 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 1392	. . 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 2786	. 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 18352	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ 𝑋))
30	2, 11, 10, 29	syl3anc 1371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ 𝑋))
31	30	oveq1d 7372	. 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 2787	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 1087   = wceq 1541   ∈ wcel 2106   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  HLchlt 37812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-psubsp 37966  df-pmap 37967  df-padd 38259

This theorem is referenced by:  dalawlem11  38344