Mathbox for Norm Megill < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddasslem12 Structured version   Visualization version   GIF version

 Description: Lemma for paddass 35914. The case when 𝑥 = 𝑦. (Contributed by NM, 11-Jan-2012.)
Hypotheses
Ref Expression
Assertion
Ref Expression
paddasslem12 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))

StepHypRef Expression
1 simpl1l 1299 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝐾 ∈ HL)
2 simpl21 1341 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑋𝐴)
3 simpl22 1343 . . . . 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 35890 . . . . 5 ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
71, 2, 3, 6syl3anc 1496 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑋 + 𝑌) ⊆ 𝐴)
8 simpl23 1345 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑍𝐴)
91, 7, 83jca 1164 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴𝑍𝐴))
104, 5sspadd2 35892 . . . 4 ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝐴) → 𝑌 ⊆ (𝑋 + 𝑌))
111, 3, 2, 10syl3anc 1496 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑌 ⊆ (𝑋 + 𝑌))
124, 5paddss1 35893 . . 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))
141hllatd 35440 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝐾 ∈ Lat)
15 simprll 799 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦𝑌)
16 simprlr 800 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑧𝑍)
17 simpl3l 1307 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝𝐴)
18 eqid 2826 . . . 4 (Base‘𝐾) = (Base‘𝐾)
19 paddasslem.l . . . 4 = (le‘𝐾)
2018, 4atbase 35365 . . . . 5 (𝑝𝐴𝑝 ∈ (Base‘𝐾))
2117, 20syl 17 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ (Base‘𝐾))
223, 15sseldd 3829 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦𝐴)
2318, 4atbase 35365 . . . . . 6 (𝑦𝐴𝑦 ∈ (Base‘𝐾))
2422, 23syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦 ∈ (Base‘𝐾))
25 simpl3r 1309 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑟𝐴)
2618, 4atbase 35365 . . . . . 6 (𝑟𝐴𝑟 ∈ (Base‘𝐾))
2725, 26syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑟 ∈ (Base‘𝐾))
28 paddasslem.j . . . . . 6 = (join‘𝐾)
2918, 28latjcl 17405 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾)) → (𝑦 𝑟) ∈ (Base‘𝐾))
3014, 24, 27, 29syl3anc 1496 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑦 𝑟) ∈ (Base‘𝐾))
318, 16sseldd 3829 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑧𝐴)
3218, 4atbase 35365 . . . . . 6 (𝑧𝐴𝑧 ∈ (Base‘𝐾))
3331, 32syl 17 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑧 ∈ (Base‘𝐾))
3418, 28latjcl 17405 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → (𝑦 𝑧) ∈ (Base‘𝐾))
3514, 24, 33, 34syl3anc 1496 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑦 𝑧) ∈ (Base‘𝐾))
36 simpl1r 1301 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑥 = 𝑦)
37 simprrl 801 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 (𝑥 𝑟))
38 oveq1 6913 . . . . . . 7 (𝑥 = 𝑦 → (𝑥 𝑟) = (𝑦 𝑟))
3938breq2d 4886 . . . . . 6 (𝑥 = 𝑦 → (𝑝 (𝑥 𝑟) ↔ 𝑝 (𝑦 𝑟)))
4039biimpa 470 . . . . 5 ((𝑥 = 𝑦𝑝 (𝑥 𝑟)) → 𝑝 (𝑦 𝑟))
4136, 37, 40syl2anc 581 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 (𝑦 𝑟))
4218, 19, 28latlej1 17414 . . . . . 6 ((𝐾 ∈ Lat ∧ 𝑦 ∈ (Base‘𝐾) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑦 (𝑦 𝑧))
4314, 24, 33, 42syl3anc 1496 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑦 (𝑦 𝑧))
44 simprrr 802 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑟 (𝑦 𝑧))
4518, 19, 28latjle12 17416 . . . . . 6 ((𝐾 ∈ Lat ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑟 ∈ (Base‘𝐾) ∧ (𝑦 𝑧) ∈ (Base‘𝐾))) → ((𝑦 (𝑦 𝑧) ∧ 𝑟 (𝑦 𝑧)) ↔ (𝑦 𝑟) (𝑦 𝑧)))
4614, 24, 27, 35, 45syl13anc 1497 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → ((𝑦 (𝑦 𝑧) ∧ 𝑟 (𝑦 𝑧)) ↔ (𝑦 𝑟) (𝑦 𝑧)))
4743, 44, 46mpbi2and 705 . . . 4 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → (𝑦 𝑟) (𝑦 𝑧))
4818, 19, 14, 21, 30, 35, 41, 47lattrd 17412 . . 3 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 (𝑦 𝑧))
4919, 28, 4, 5elpaddri 35878 . . 3 (((𝐾 ∈ Lat ∧ 𝑌𝐴𝑍𝐴) ∧ (𝑦𝑌𝑧𝑍) ∧ (𝑝𝐴𝑝 (𝑦 𝑧))) → 𝑝 ∈ (𝑌 + 𝑍))
5014, 3, 8, 15, 16, 17, 48, 49syl322anc 1523 . 2 ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ (𝑌 + 𝑍))
5113, 50sseldd 3829 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 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166   ⊆ wss 3799   class class class wbr 4874  ‘cfv 6124  (class class class)co 6906  Basecbs 16223  lecple 16313  joincjn 17298  Latclat 17399  Atomscatm 35339  HLchlt 35426  +𝑃cpadd 35871 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-poset 17300  df-lub 17328  df-glb 17329  df-join 17330  df-meet 17331  df-lat 17400  df-ats 35343  df-atl 35374  df-cvlat 35398  df-hlat 35427  df-padd 35872 This theorem is referenced by:  paddasslem14  35909
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