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Theorem pl42lem3N 35782
Description: Lemma for pl42N 35784. (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 1226 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝐾 ∈ HL)
2 simpl2 1228 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑋𝐵)
3 pl42lem.b . . . . . 6 𝐵 = (Base‘𝐾)
4 eqid 2770 . . . . . 6 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pl42lem.f . . . . . 6 𝐹 = (pmap‘𝐾)
63, 4, 5pmapssat 35560 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
71, 2, 6syl2anc 565 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑋) ⊆ (Atoms‘𝐾))
8 simpl3 1230 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑌𝐵)
93, 4, 5pmapssat 35560 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝐵) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
101, 8, 9syl2anc 565 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑌) ⊆ (Atoms‘𝐾))
11 pl42lem.p . . . . 5 + = (+𝑃𝐾)
124, 11paddssat 35615 . . . 4 ((𝐾 ∈ HL ∧ (𝐹𝑋) ⊆ (Atoms‘𝐾) ∧ (𝐹𝑌) ⊆ (Atoms‘𝐾)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
131, 7, 10, 12syl3anc 1475 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾))
14 simpr2 1234 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑊𝐵)
153, 4, 5pmapssat 35560 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
161, 14, 15syl2anc 565 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑊) ⊆ (Atoms‘𝐾))
17 inss1 3979 . . . 4 (((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) ⊆ ((𝐹𝑋) + (𝐹𝑌))
184, 11paddss1 35618 . . . 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 1475 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)))
21 simpr3 1236 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → 𝑉𝐵)
223, 4, 5pmapssat 35560 . . . 4 ((𝐾 ∈ HL ∧ 𝑉𝐵) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
231, 21, 22syl2anc 565 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (Atoms‘𝐾))
244, 11sspadd2 35617 . . 3 ((𝐾 ∈ HL ∧ (𝐹𝑉) ⊆ (Atoms‘𝐾) ∧ ((𝐹𝑋) + (𝐹𝑌)) ⊆ (Atoms‘𝐾)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
251, 23, 13, 24syl3anc 1475 . 2 (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (𝐹𝑉) ⊆ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉)))
26 ss2in 3987 . 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 565 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 1070   = wceq 1630  wcel 2144  cin 3720  wss 3721  cfv 6031  (class class class)co 6792  Basecbs 16063  lecple 16155  occoc 16156  joincjn 17151  meetcmee 17152  Atomscatm 35065  HLchlt 35152  pmapcpmap 35298  +𝑃cpadd 35596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-pmap 35305  df-padd 35597
This theorem is referenced by:  pl42lem4N  35783
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