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Theorem paddasslem13 36967
Description: Lemma for paddass 36973. The case when 𝑟 (𝑥 𝑦). (Unlike the proof in Maeda and Maeda, we don't need 𝑥𝑦.) (Contributed by NM, 11-Jan-2012.)
Hypotheses
Ref Expression
paddasslem.l = (le‘𝐾)
paddasslem.j = (join‘𝐾)
paddasslem.a 𝐴 = (Atoms‘𝐾)
paddasslem.p + = (+𝑃𝐾)
Assertion
Ref Expression
paddasslem13 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Proof of Theorem paddasslem13
StepHypRef Expression
1 simpl1l 1220 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝐾 ∈ HL)
2 simpl21 1247 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑋𝐴)
3 simpl22 1248 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑌𝐴)
4 paddasslem.a . . . . 5 𝐴 = (Atoms‘𝐾)
5 paddasslem.p . . . . 5 + = (+𝑃𝐾)
64, 5paddssat 36949 . . . 4 ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
71, 2, 3, 6syl3anc 1367 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → (𝑋 + 𝑌) ⊆ 𝐴)
8 simpl23 1249 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑍𝐴)
94, 5sspadd1 36950 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴𝑍𝐴) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
101, 7, 8, 9syl3anc 1367 . 2 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
111hllatd 36499 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝐾 ∈ Lat)
12 simprll 777 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑥𝑋)
13 simprlr 778 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑦𝑌)
14 simpl3l 1224 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝𝐴)
15 eqid 2821 . . . 4 (Base‘𝐾) = (Base‘𝐾)
16 paddasslem.l . . . 4 = (le‘𝐾)
1715, 4atbase 36424 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
1814, 17syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 ∈ (Base‘𝐾))
192, 12sseldd 3967 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑥𝐴)
2015, 4atbase 36424 . . . . . 6 (𝑥𝐴𝑥 ∈ (Base‘𝐾))
2119, 20syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑥 ∈ (Base‘𝐾))
22 simpl3r 1225 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑟𝐴)
2315, 4atbase 36424 . . . . . 6 (𝑟𝐴𝑟 ∈ (Base‘𝐾))
2422, 23syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑟 ∈ (Base‘𝐾))
25 paddasslem.j . . . . . 6 = (join‘𝐾)
2615, 25latjcl 17660 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑥 𝑟) ∈ (Base‘𝐾))
2711, 21, 24, 26syl3anc 1367 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → (𝑥 𝑟) ∈ (Base‘𝐾))
283, 13sseldd 3967 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑦𝐴)
2915, 4atbase 36424 . . . . . 6 (𝑦𝐴𝑦 ∈ (Base‘𝐾))
3028, 29syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑦 ∈ (Base‘𝐾))
3115, 25latjcl 17660 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥 𝑦) ∈ (Base‘𝐾))
3211, 21, 30, 31syl3anc 1367 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → (𝑥 𝑦) ∈ (Base‘𝐾))
33 simprrr 780 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 (𝑥 𝑟))
3415, 16, 25latlej1 17669 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → 𝑥 (𝑥 𝑦))
3511, 21, 30, 34syl3anc 1367 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑥 (𝑥 𝑦))
36 simprrl 779 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑟 (𝑥 𝑦))
3715, 16, 25latjle12 17671 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (𝑥 𝑦) ∈ (Base‘𝐾))) → ((𝑥 (𝑥 𝑦) ∧ 𝑟 (𝑥 𝑦)) ↔ (𝑥 𝑟) (𝑥 𝑦)))
3811, 21, 24, 32, 37syl13anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → ((𝑥 (𝑥 𝑦) ∧ 𝑟 (𝑥 𝑦)) ↔ (𝑥 𝑟) (𝑥 𝑦)))
3935, 36, 38mpbi2and 710 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → (𝑥 𝑟) (𝑥 𝑦))
4015, 16, 11, 18, 27, 32, 33, 39lattrd 17667 . . 3 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 (𝑥 𝑦))
4116, 25, 4, 5elpaddri 36937 . . 3 (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑥𝑋𝑦𝑌) ∧ (𝑝𝐴𝑝 (𝑥 𝑦))) → 𝑝 ∈ (𝑋 + 𝑌))
4211, 2, 3, 12, 13, 14, 40, 41syl322anc 1394 . 2 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 ∈ (𝑋 + 𝑌))
4310, 42sseldd 3967 1 ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wss 3935   class class class wbr 5065  cfv 6354  (class class class)co 7155  Basecbs 16482  lecple 16571  joincjn 17553  Latclat 17654  Atomscatm 36398  HLchlt 36485  +𝑃cpadd 36930
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-poset 17555  df-lub 17583  df-glb 17584  df-join 17585  df-meet 17586  df-lat 17655  df-ats 36402  df-atl 36433  df-cvlat 36457  df-hlat 36486  df-padd 36931
This theorem is referenced by:  paddasslem14  36968
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