Theorem pmapjlln1 40553

Description: The projective map of the join of a lattice element and a lattice line (expressed as the join 𝑄 ∨ 𝑅 of two atoms). (Contributed by NM, 16-Sep-2012.)

Hypotheses

Ref	Expression
pmapjat.b	⊢ 𝐵 = (Base‘𝐾)
pmapjat.j	⊢ ∨ = (join‘𝐾)
pmapjat.a	⊢ 𝐴 = (Atoms‘𝐾)
pmapjat.m	⊢ 𝑀 = (pmap‘𝐾)
pmapjat.p	⊢ + = (+𝑃‘𝐾)

Assertion

Ref	Expression
pmapjlln1	⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((𝑀‘𝑋) + (𝑀‘(𝑄 ∨ 𝑅))))

Proof of Theorem pmapjlln1

Step	Hyp	Ref	Expression
1		simpl 487	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ HL)
2		pmapjat.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		pmapjat.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		pmapjat.m	. . . . 5 ⊢ 𝑀 = (pmap‘𝐾)
5	2, 3, 4	pmapssat 40457	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴)
6	5	3ad2antr1 1205	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘𝑋) ⊆ 𝐴)
7		simpr2 1212	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
8	2, 3	atbase 39987	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
9	7, 8	syl 18	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
10	2, 3, 4	pmapssat 40457	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵) → (𝑀‘𝑄) ⊆ 𝐴)
11	9, 10	syldan 602	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘𝑄) ⊆ 𝐴)
12		simpr3 1213	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐴)
13	2, 3	atbase 39987	. . . . 5 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
14	12, 13	syl 18	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐵)
15	2, 3, 4	pmapssat 40457	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐵) → (𝑀‘𝑅) ⊆ 𝐴)
16	14, 15	syldan 602	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘𝑅) ⊆ 𝐴)
17		pmapjat.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
18	3, 17	paddass 40536	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((𝑀‘𝑋) ⊆ 𝐴 ∧ (𝑀‘𝑄) ⊆ 𝐴 ∧ (𝑀‘𝑅) ⊆ 𝐴)) → (((𝑀‘𝑋) + (𝑀‘𝑄)) + (𝑀‘𝑅)) = ((𝑀‘𝑋) + ((𝑀‘𝑄) + (𝑀‘𝑅))))
19	1, 6, 11, 16, 18	syl13anc 1397	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (((𝑀‘𝑋) + (𝑀‘𝑄)) + (𝑀‘𝑅)) = ((𝑀‘𝑋) + ((𝑀‘𝑄) + (𝑀‘𝑅))))
20		hllat 40061	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
21	20	adantr 485	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ Lat)
22		simpr1 1211	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
23		pmapjat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
24	2, 23	latjcl 18495	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
25	21, 22, 9, 24	syl3anc 1396	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
26	2, 23, 3, 4, 17	pmapjat1 40551	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑄) ∈ 𝐵 ∧ 𝑅 ∈ 𝐴) → (𝑀‘((𝑋 ∨ 𝑄) ∨ 𝑅)) = ((𝑀‘(𝑋 ∨ 𝑄)) + (𝑀‘𝑅)))
27	1, 25, 12, 26	syl3anc 1396	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘((𝑋 ∨ 𝑄) ∨ 𝑅)) = ((𝑀‘(𝑋 ∨ 𝑄)) + (𝑀‘𝑅)))
28	2, 23	latjass 18539	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑅 ∈ 𝐵)) → ((𝑋 ∨ 𝑄) ∨ 𝑅) = (𝑋 ∨ (𝑄 ∨ 𝑅)))
29	21, 22, 9, 14, 28	syl13anc 1397	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ 𝑅) = (𝑋 ∨ (𝑄 ∨ 𝑅)))
30	29	fveq2d 6886	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘((𝑋 ∨ 𝑄) ∨ 𝑅)) = (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))))
31	2, 23, 3, 4, 17	pmapjat1 40551	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
32	31	3adant3r3 1201	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
33	32	oveq1d 7426	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑀‘(𝑋 ∨ 𝑄)) + (𝑀‘𝑅)) = (((𝑀‘𝑋) + (𝑀‘𝑄)) + (𝑀‘𝑅)))
34	27, 30, 33	3eqtr3d 2812	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))) = (((𝑀‘𝑋) + (𝑀‘𝑄)) + (𝑀‘𝑅)))
35	2, 23, 3, 4, 17	pmapjat1 40551	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵 ∧ 𝑅 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑅)) = ((𝑀‘𝑄) + (𝑀‘𝑅)))
36	1, 9, 12, 35	syl3anc 1396	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑄 ∨ 𝑅)) = ((𝑀‘𝑄) + (𝑀‘𝑅)))
37	36	oveq2d 7427	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑀‘𝑋) + (𝑀‘(𝑄 ∨ 𝑅))) = ((𝑀‘𝑋) + ((𝑀‘𝑄) + (𝑀‘𝑅))))
38	19, 34, 37	3eqtr4d 2814	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((𝑀‘𝑋) + (𝑀‘(𝑄 ∨ 𝑅))))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  joincjn 18367  Latclat 18487  Atomscatm 39961  HLchlt 40048  pmapcpmap 40195  +_𝑃cpadd 40493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-lat 18488  df-clat 18555  df-oposet 39874  df-ol 39876  df-oml 39877  df-covers 39964  df-ats 39965  df-atl 39996  df-cvlat 40020  df-hlat 40049  df-pmap 40202  df-padd 40494

This theorem is referenced by:  llnmod1i2  40558