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Theorem pl42lem3N 39964
Description: Lemma for pl42N 39966. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pl42lem.b 𝐵 = (Base‘𝐾)
pl42lem.l = (le‘𝐾)
pl42lem.j = (join‘𝐾)
pl42lem.m = (meet‘𝐾)
pl42lem.o = (oc‘𝐾)
pl42lem.f 𝐹 = (pmap‘𝐾)
pl42lem.p + = (+𝑃𝐾)
Assertion
Ref Expression
pl42lem3N (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉)) ⊆ ((((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)) ∩ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉))))

Proof of Theorem pl42lem3N
StepHypRef Expression
1 simpl1 1190 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
2 simpl2 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
3 pl42lem.b . . . . . 6 𝐵 = (Base‘𝐾)
4 eqid 2735 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
63, 4, 5pmapssat 39742 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
71, 2, 6syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
8 simpl3 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
93, 4, 5pmapssat 39742 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
101, 8, 9syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
11 pl42lem.p . . . . 5 + = (+𝑃𝐾)
124, 11paddssat 39797 . . . 4 ((𝐾 ∈ HL ∧ (𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
131, 7, 10, 12syl3anc 1370 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
14 simpr2 1194 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
153, 4, 5pmapssat 39742 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
161, 14, 15syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
17 inss1 4245 . . . 4 (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) ⊆ ((𝐹𝑋) + (𝐹𝑌))
184, 11paddss1 39800 . . . 4 ((𝐾 ∈ HL ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑊) ⊆ (Atoms‘𝐾)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) ⊆ ((𝐹𝑋) + (𝐹𝑌)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊))))
1917, 18mpi 20 . . 3 ((𝐾 ∈ HL ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑊) ⊆ (Atoms‘𝐾)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)))
201, 13, 16, 19syl3anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)))
21 simpr3 1195 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
223, 4, 5pmapssat 39742 . . . 4 ((𝐾 ∈ HL ∧ 𝑉𝐵) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
244, 11sspadd2 39799 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑉) ⊆ (Atoms‘𝐾) ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
251, 23, 13, 24syl3anc 1370 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
26 ss2in 4253 . 2 ((((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)) ∧ (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉))) → (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉)) ⊆ ((((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)) ∩ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉))))
2720, 25, 26syl2anc 584 1 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉)) ⊆ ((((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)) ∩ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963  cfv 6563  (class class class)co 7431  Basecbs 17245  lecple 17305  occoc 17306  joincjn 18369  meetcmee 18370  Atomscatm 39245  HLchlt 39332  pmapcpmap 39480  +𝑃cpadd 39778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-pmap 39487  df-padd 39779
This theorem is referenced by:  pl42lem4N  39965
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