Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  paddasslem11 Structured version   Visualization version   GIF version
Theorem paddasslem11 39227
Description: Lemma for paddass 39235. The case when 𝑝 = 𝑧. (Contributed by NM, 11-Jan-2012.)
Hypotheses
Ref Expression
paddasslem.l ≀ = (leβ€˜πΎ)
paddasslem.j ∨ = (joinβ€˜πΎ)
paddasslem.a 𝐴 = (Atomsβ€˜πΎ)
paddasslem.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
paddasslem11 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
Proof of Theorem paddasslem11
StepHypRef Expression
1 simplll 774 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝐾 ∈ HL)
2 simplr3 1215 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑍 βŠ† 𝐴)
3 simplr1 1213 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑋 βŠ† 𝐴)
4 simplr2 1214 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ π‘Œ βŠ† 𝐴)
5 paddasslem.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
6 paddasslem.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
75, 6paddssat 39211 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) βŠ† 𝐴)
81, 3, 4, 7syl3anc 1369 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ (𝑋 + π‘Œ) βŠ† 𝐴)
95, 6sspadd2 39213 . . 3 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ (𝑋 + π‘Œ) βŠ† 𝐴) β†’ 𝑍 βŠ† ((𝑋 + π‘Œ) + 𝑍))
101, 2, 8, 9syl3anc 1369 . 2 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑍 βŠ† ((𝑋 + π‘Œ) + 𝑍))
11 simpllr 775 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 = 𝑧)
12 simpr 484 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑧 ∈ 𝑍)
1311, 12eqeltrd 2828 . 2 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 ∈ 𝑍)
1410, 13sseldd 3979 1 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   βŠ† wss 3944  β€˜cfv 6542  (class class class)co 7414  lecple 17225  joincjn 18288  Atomscatm 38659  HLchlt 38746  +𝑃cpadd 39192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-padd 39193
This theorem is referenced by:  paddasslem14  39230
  Copyright terms: Public domain W3C validator