Theorem llnmod2i2 40123

Description: Version of modular law pmod1i 40108 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 1204	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ HL)
2	1	hllatd 39624	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝐾 ∈ Lat)
3		simp13 1206	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑌 ∈ 𝐵)
4		simp2l 1200	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → 𝑃 ∈ 𝐴)
5		simp2r 1201	. . . . 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 39627	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ 𝐵)
10	1, 4, 5, 9	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑃 ∨ 𝑄) ∈ 𝐵)
11		simp12 1205	. . . 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 18363	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵)
14	2, 10, 11, 13	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵)
15	6, 7	latjcom 18370	. . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ ((𝑃 ∨ 𝑄) ∧ 𝑋) ∈ 𝐵) → (𝑌 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑋)) = (((𝑃 ∨ 𝑄) ∧ 𝑋) ∨ 𝑌))
16	2, 3, 14, 15	syl3anc 1373	. 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 18362	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
18	2, 3, 10, 17	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵)
19	6, 12	latmcom 18386	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄)) ∈ 𝐵) → (𝑋 ∧ (𝑌 ∨ (𝑃 ∨ 𝑄))) = ((𝑌 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑋))
20	2, 11, 18, 19	syl3anc 1373	. . 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 18370	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∨ 𝑄) ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑃 ∨ 𝑄) ∨ 𝑌) = (𝑌 ∨ (𝑃 ∨ 𝑄)))
22	2, 10, 3, 21	syl3anc 1373	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑃 ∨ 𝑄) ∨ 𝑌) = (𝑌 ∨ (𝑃 ∨ 𝑄)))
23	22	oveq2d 7374	. . 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 40120	. . . 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 1394	. . 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 2781	. 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 18386	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ (𝑃 ∨ 𝑄) ∈ 𝐵) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ 𝑋))
30	2, 11, 10, 29	syl3anc 1373	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → (𝑋 ∧ (𝑃 ∨ 𝑄)) = ((𝑃 ∨ 𝑄) ∧ 𝑋))
31	30	oveq1d 7373	. 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 2782	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 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   class class class wbr 5098  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  lecple 17184  joincjn 18234  meetcmee 18235  Latclat 18354  Atomscatm 39523  HLchlt 39610

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-lat 18355  df-clat 18422  df-oposet 39436  df-ol 39438  df-oml 39439  df-covers 39526  df-ats 39527  df-atl 39558  df-cvlat 39582  df-hlat 39611  df-psubsp 39763  df-pmap 39764  df-padd 40056

This theorem is referenced by:  dalawlem11  40141