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Theorem pl42lem3N 34068
Description: Lemma for pl42N 34070. (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 1056 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
2 simpl2 1057 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
3 pl42lem.b . . . . . 6 𝐵 = (Base‘𝐾)
4 eqid 2609 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
63, 4, 5pmapssat 33846 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
71, 2, 6syl2anc 690 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
8 simpl3 1058 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
93, 4, 5pmapssat 33846 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
101, 8, 9syl2anc 690 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
11 pl42lem.p . . . . 5 + = (+𝑃𝐾)
124, 11paddssat 33901 . . . 4 ((𝐾 ∈ HL ∧ (𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
131, 7, 10, 12syl3anc 1317 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
14 simpr2 1060 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
153, 4, 5pmapssat 33846 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
161, 14, 15syl2anc 690 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
17 inss1 3794 . . . 4 (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) ⊆ ((𝐹𝑋) + (𝐹𝑌))
184, 11paddss1 33904 . . . 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 1317 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)))
21 simpr3 1061 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
223, 4, 5pmapssat 33846 . . . 4 ((𝐾 ∈ HL ∧ 𝑉𝐵) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 690 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
244, 11sspadd2 33903 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑉) ⊆ (Atoms‘𝐾) ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
251, 23, 13, 24syl3anc 1317 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
26 ss2in 3801 . 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 690 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 382  w3a 1030   = wceq 1474  wcel 1976  cin 3538  wss 3539  cfv 5789  (class class class)co 6526  Basecbs 15643  lecple 15723  occoc 15724  joincjn 16715  meetcmee 16716  Atomscatm 33351  HLchlt 33438  pmapcpmap 33584  +𝑃cpadd 33882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-pmap 33591  df-padd 33883
This theorem is referenced by:  pl42lem4N  34069
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