pmapjat2 - Mathbox for Norm Megill

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Theorem pmapjat2 38317

Description: The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-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
pmapjat2	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋)))

**Proof of Theorem pmapjat2**
Step	Hyp	Ref	Expression
1		pmapjat.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		pmapjat.j	. . 3 ⊢ ∨ = (join‘𝐾)
3		pmapjat.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
4		pmapjat.m	. . 3 ⊢ 𝑀 = (pmap‘𝐾)
5		pmapjat.p	. . 3 ⊢ + = (+_𝑃‘𝐾)
6	1, 2, 3, 4, 5	pmapjat1 38316	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
7		hllat 37825	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
8	7	3ad2ant1 1133	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Lat)
9	1, 3	atbase 37751	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
10	9	3ad2ant3 1135	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐵)
11		simp2 1137	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑋 ∈ 𝐵)
12	1, 2	latjcom 18336	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
13	8, 10, 11, 12	syl3anc 1371	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
14	13	fveq2d 6846	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = (𝑀‘(𝑋 ∨ 𝑄)))
15		simp1 1136	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ HL)
16	1, 3, 4	pmapssat 38222	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵) → (𝑀‘𝑄) ⊆ 𝐴)
17	15, 10, 16	syl2anc 584	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑄) ⊆ 𝐴)
18	1, 3, 4	pmapssat 38222	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴)
19	18	3adant3 1132	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑋) ⊆ 𝐴)
20	3, 5	paddcom 38276	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑀‘𝑄) ⊆ 𝐴 ∧ (𝑀‘𝑋) ⊆ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
21	8, 17, 19, 20	syl3anc 1371	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
22	6, 14, 21	3eqtr4d 2786	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 joincjn 18200 Latclat 18320 Atomscatm 37725 HLchlt 37812 pmapcpmap 37960 +_𝑃cpadd 38258

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-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-pmap 37967 df-padd 38259

This theorem is referenced by: atmod1i1 38320

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