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Theorem pl42lem3N 36986
Description: Lemma for pl42N 36988. (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 1185 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
2 simpl2 1186 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
3 pl42lem.b . . . . . 6 𝐵 = (Base‘𝐾)
4 eqid 2825 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
63, 4, 5pmapssat 36764 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
71, 2, 6syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
8 simpl3 1187 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
93, 4, 5pmapssat 36764 . . . . 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 36819 . . . 4 ((𝐾 ∈ HL ∧ (𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
131, 7, 10, 12syl3anc 1365 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
14 simpr2 1189 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
153, 4, 5pmapssat 36764 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
161, 14, 15syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
17 inss1 4208 . . . 4 (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) ⊆ ((𝐹𝑋) + (𝐹𝑌))
184, 11paddss1 36822 . . . 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 1365 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)))
21 simpr3 1190 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
223, 4, 5pmapssat 36764 . . . 4 ((𝐾 ∈ HL ∧ 𝑉𝐵) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
244, 11sspadd2 36821 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑉) ⊆ (Atoms‘𝐾) ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
251, 23, 13, 24syl3anc 1365 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
26 ss2in 4216 . 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 396  w3a 1081   = wceq 1530  wcel 2107  cin 3938  wss 3939  cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  occoc 16565  joincjn 17546  meetcmee 17547  Atomscatm 36268  HLchlt 36355  pmapcpmap 36502  +𝑃cpadd 36800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-pmap 36509  df-padd 36801
This theorem is referenced by:  pl42lem4N  36987
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