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Theorem pl42lem3N 39984
Description: Lemma for pl42N 39986. (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 1191 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
2 simpl2 1192 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
3 pl42lem.b . . . . . 6 𝐵 = (Base‘𝐾)
4 eqid 2736 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
63, 4, 5pmapssat 39762 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
71, 2, 6syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
8 simpl3 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
93, 4, 5pmapssat 39762 . . . . 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 39817 . . . 4 ((𝐾 ∈ HL ∧ (𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
131, 7, 10, 12syl3anc 1372 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
14 simpr2 1195 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
153, 4, 5pmapssat 39762 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
161, 14, 15syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
17 inss1 4236 . . . 4 (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) ⊆ ((𝐹𝑋) + (𝐹𝑌))
184, 11paddss1 39820 . . . 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 1372 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)))
21 simpr3 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
223, 4, 5pmapssat 39762 . . . 4 ((𝐾 ∈ HL ∧ 𝑉𝐵) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
244, 11sspadd2 39819 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑉) ⊆ (Atoms‘𝐾) ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
251, 23, 13, 24syl3anc 1372 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
26 ss2in 4244 . 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 1539  wcel 2107  cin 3949  wss 3950  cfv 6560  (class class class)co 7432  Basecbs 17248  lecple 17305  occoc 17306  joincjn 18358  meetcmee 18359  Atomscatm 39265  HLchlt 39352  pmapcpmap 39500  +𝑃cpadd 39798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-pmap 39507  df-padd 39799
This theorem is referenced by:  pl42lem4N  39985
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