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Theorem paddasslem12 35435
Description: Lemma for paddass 35442. The case when 𝑥 = 𝑦. (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
paddasslem12 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

Proof of Theorem paddasslem12
StepHypRef Expression
1 simpl1l 1132 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝐾 ∈ HL)
2 simpl21 1159 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑋𝐴)
3 simpl22 1160 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑌𝐴)
4 paddasslem.a . . . . . 6 𝐴 = (Atoms‘𝐾)
5 paddasslem.p . . . . . 6 + = (+𝑃𝐾)
64, 5paddssat 35418 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
71, 2, 3, 6syl3anc 1366 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑋 + 𝑌) ⊆ 𝐴)
8 simpl23 1161 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑍𝐴)
91, 7, 83jca 1261 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴𝑍𝐴))
104, 5sspadd2 35420 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝐴) → 𝑌 ⊆ (𝑋 + 𝑌))
111, 3, 2, 10syl3anc 1366 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑌 ⊆ (𝑋 + 𝑌))
124, 5paddss1 35421 . . 3 ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴𝑍𝐴) → (𝑌 ⊆ (𝑋 + 𝑌) → (𝑌 + 𝑍) ⊆ ((𝑋 + 𝑌) + 𝑍)))
139, 11, 12sylc 65 . 2 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑌 + 𝑍) ⊆ ((𝑋 + 𝑌) + 𝑍))
14 hllat 34968 . . . 4 (𝐾 ∈ HL → 𝐾 ∈ Lat)
151, 14syl 17 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝐾 ∈ Lat)
16 simprll 819 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦𝑌)
17 simprlr 820 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑧𝑍)
18 simpl3l 1136 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝𝐴)
19 eqid 2651 . . . 4 (Base‘𝐾) = (Base‘𝐾)
20 paddasslem.l . . . 4 = (le‘𝐾)
2119, 4atbase 34894 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
2218, 21syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ (Base‘𝐾))
233, 16sseldd 3637 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦𝐴)
2419, 4atbase 34894 . . . . . 6 (𝑦𝐴𝑦 ∈ (Base‘𝐾))
2523, 24syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦 ∈ (Base‘𝐾))
26 simpl3r 1137 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑟𝐴)
2719, 4atbase 34894 . . . . . 6 (𝑟𝐴𝑟 ∈ (Base‘𝐾))
2826, 27syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑟 ∈ (Base‘𝐾))
29 paddasslem.j . . . . . 6 = (join‘𝐾)
3019, 29latjcl 17098 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑦 𝑟) ∈ (Base‘𝐾))
3115, 25, 28, 30syl3anc 1366 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑦 𝑟) ∈ (Base‘𝐾))
328, 17sseldd 3637 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑧𝐴)
3319, 4atbase 34894 . . . . . 6 (𝑧𝐴𝑧 ∈ (Base‘𝐾))
3432, 33syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑧 ∈ (Base‘𝐾))
3519, 29latjcl 17098 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑦 𝑧) ∈ (Base‘𝐾))
3615, 25, 34, 35syl3anc 1366 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑦 𝑧) ∈ (Base‘𝐾))
37 simpl1r 1133 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑥 = 𝑦)
38 simprrl 821 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 (𝑥 𝑟))
39 oveq1 6697 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 𝑟) = (𝑦 𝑟))
4039breq2d 4697 . . . . . 6 (𝑥 = 𝑦 → (𝑝 (𝑥 𝑟) ↔ 𝑝 (𝑦 𝑟)))
4140biimpa 500 . . . . 5 ((𝑥 = 𝑦𝑝 (𝑥 𝑟)) → 𝑝 (𝑦 𝑟))
4237, 38, 41syl2anc 694 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 (𝑦 𝑟))
4319, 20, 29latlej1 17107 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑦 (𝑦 𝑧))
4415, 25, 34, 43syl3anc 1366 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦 (𝑦 𝑧))
45 simprrr 822 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑟 (𝑦 𝑧))
4619, 20, 29latjle12 17109 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (𝑦 𝑧) ∈ (Base‘𝐾))) → ((𝑦 (𝑦 𝑧) ∧ 𝑟 (𝑦 𝑧)) ↔ (𝑦 𝑟) (𝑦 𝑧)))
4715, 25, 28, 36, 46syl13anc 1368 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → ((𝑦 (𝑦 𝑧) ∧ 𝑟 (𝑦 𝑧)) ↔ (𝑦 𝑟) (𝑦 𝑧)))
4844, 45, 47mpbi2and 976 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑦 𝑟) (𝑦 𝑧))
4919, 20, 15, 22, 31, 36, 42, 48lattrd 17105 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 (𝑦 𝑧))
5020, 29, 4, 5elpaddri 35406 . . 3 (((𝐾 ∈ Lat ∧ 𝑌𝐴𝑍𝐴) ∧ (𝑦𝑌𝑧𝑍) ∧ (𝑝𝐴𝑝 (𝑦 𝑧))) → 𝑝 ∈ (𝑌 + 𝑍))
5115, 3, 8, 16, 17, 18, 49, 50syl322anc 1394 . 2 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ (𝑌 + 𝑍))
5213, 51sseldd 3637 1 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607   class class class wbr 4685  cfv 5926  (class class class)co 6690  Basecbs 15904  lecple 15995  joincjn 16991  Latclat 17092  Atomscatm 34868  HLchlt 34955  +𝑃cpadd 35399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-poset 16993  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-lat 17093  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-padd 35400
This theorem is referenced by:  paddasslem14  35437
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