pmapjat2 - Mathbox for Norm Megill

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Theorem pmapjat2 37795

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 37794	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
7		hllat 37304	. . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
8	7	3ad2ant1 1131	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ Lat)
9	1, 3	atbase 37230	. . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵)
10	9	3ad2ant3 1133	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐵)
11		simp2 1135	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝑋 ∈ 𝐵)
12	1, 2	latjcom 18080	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
13	8, 10, 11, 12	syl3anc 1369	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑄 ∨ 𝑋) = (𝑋 ∨ 𝑄))
14	13	fveq2d 6760	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = (𝑀‘(𝑋 ∨ 𝑄)))
15		simp1 1134	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → 𝐾 ∈ HL)
16	1, 3, 4	pmapssat 37700	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐵) → (𝑀‘𝑄) ⊆ 𝐴)
17	15, 10, 16	syl2anc 583	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑄) ⊆ 𝐴)
18	1, 3, 4	pmapssat 37700	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴)
19	18	3adant3 1130	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘𝑋) ⊆ 𝐴)
20	3, 5	paddcom 37754	. . 3 ⊢ ((𝐾 ∈ Lat ∧ (𝑀‘𝑄) ⊆ 𝐴 ∧ (𝑀‘𝑋) ⊆ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
21	8, 17, 19, 20	syl3anc 1369	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → ((𝑀‘𝑄) + (𝑀‘𝑋)) = ((𝑀‘𝑋) + (𝑀‘𝑄)))
22	6, 14, 21	3eqtr4d 2788	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 1085 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 joincjn 17944 Latclat 18064 Atomscatm 37204 HLchlt 37291 pmapcpmap 37438 +_𝑃cpadd 37736

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-lat 18065 df-clat 18132 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-pmap 37445 df-padd 37737

This theorem is referenced by: atmod1i1 37798

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