Theorem pmapjlln1 40480

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 486	. . 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 40384	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴)
6	5	3ad2antr1 1203	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘𝑋) ⊆ 𝐴)
7		simpr2 1210	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑄 ∈ 𝐴)
8	2, 3	atbase 39914	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
9	7, 8	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑄 ∈ 𝐵)
10	2, 3, 4	pmapssat 40384	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵) → (𝑀‘𝑄) ⊆ 𝐴)
11	9, 10	syldan 600	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘𝑄) ⊆ 𝐴)
12		simpr3 1211	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐴)
13	2, 3	atbase 39914	. . . . 5 ⊢ (𝑅 ∈ 𝐴 → 𝑅 ∈ 𝐵)
14	12, 13	syl 17	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑅 ∈ 𝐵)
15	2, 3, 4	pmapssat 40384	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐵) → (𝑀‘𝑅) ⊆ 𝐴)
16	14, 15	syldan 600	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘𝑅) ⊆ 𝐴)
17		pmapjat.p	. . . 4 ⊢ + = (+𝑃‘𝐾)
18	3, 17	paddass 40463	. . 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 1392	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (((𝑀‘𝑋) + (𝑀‘𝑄)) + (𝑀‘𝑅)) = ((𝑀‘𝑋) + ((𝑀‘𝑄) + (𝑀‘𝑅))))
20		hllat 39988	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
21	20	adantr 484	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝐾 ∈ Lat)
22		simpr1 1209	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → 𝑋 ∈ 𝐵)
23		pmapjat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
24	2, 23	latjcl 18472	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑋 ∨ 𝑄) ∈ 𝐵)
25	21, 22, 9, 24	syl3anc 1391	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑋 ∨ 𝑄) ∈ 𝐵)
26	2, 23, 3, 4, 17	pmapjat1 40478	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∨ 𝑄) ∈ 𝐵 ∧ 𝑅 ∈ 𝐴) → (𝑀‘((𝑋 ∨ 𝑄) ∨ 𝑅)) = ((𝑀‘(𝑋 ∨ 𝑄)) + (𝑀‘𝑅)))
27	1, 25, 12, 26	syl3anc 1391	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘((𝑋 ∨ 𝑄) ∨ 𝑅)) = ((𝑀‘(𝑋 ∨ 𝑄)) + (𝑀‘𝑅)))
28	2, 23	latjass 18516	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵 ∧ 𝑅 ∈ 𝐵)) → ((𝑋 ∨ 𝑄) ∨ 𝑅) = (𝑋 ∨ (𝑄 ∨ 𝑅)))
29	21, 22, 9, 14, 28	syl13anc 1392	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑋 ∨ 𝑄) ∨ 𝑅) = (𝑋 ∨ (𝑄 ∨ 𝑅)))
30	29	fveq2d 6872	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘((𝑋 ∨ 𝑄) ∨ 𝑅)) = (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))))
31	2, 23, 3, 4, 17	pmapjat1 40478	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
32	31	3adant3r3 1199	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
33	32	oveq1d 7412	. . 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 2806	. 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 40478	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵 ∧ 𝑅 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑅)) = ((𝑀‘𝑄) + (𝑀‘𝑅)))
36	1, 9, 12, 35	syl3anc 1391	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑄 ∨ 𝑅)) = ((𝑀‘𝑄) + (𝑀‘𝑅)))
37	36	oveq2d 7413	. 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 2808	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((𝑀‘𝑋) + (𝑀‘(𝑄 ∨ 𝑅))))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ⊆ wss 3905  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  joincjn 18344  Latclat 18464  Atomscatm 39888  HLchlt 39975  pmapcpmap 40122  +_𝑃cpadd 40420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-riota 7354  df-ov 7400  df-oprab 7401  df-mpo 7402  df-1st 7971  df-2nd 7972  df-proset 18327  df-poset 18346  df-plt 18361  df-lub 18377  df-glb 18378  df-join 18379  df-meet 18380  df-p0 18456  df-lat 18465  df-clat 18532  df-oposet 39801  df-ol 39803  df-oml 39804  df-covers 39891  df-ats 39892  df-atl 39923  df-cvlat 39947  df-hlat 39976  df-pmap 40129  df-padd 40421

This theorem is referenced by:  llnmod1i2  40485