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Theorem paddasslem11 39355
Description: Lemma for paddass 39363. 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 773 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝐾 ∈ HL)
2 simplr3 1214 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑍 βŠ† 𝐴)
3 simplr1 1212 . . . 4 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑋 βŠ† 𝐴)
4 simplr2 1213 . . . 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 39339 . . . 4 ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 + π‘Œ) βŠ† 𝐴)
81, 3, 4, 7syl3anc 1368 . . 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 39341 . . 3 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ (𝑋 + π‘Œ) βŠ† 𝐴) β†’ 𝑍 βŠ† ((𝑋 + π‘Œ) + 𝑍))
101, 2, 8, 9syl3anc 1368 . 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 774 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 = 𝑧)
12 simpr 483 . . 3 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑧 ∈ 𝑍)
1311, 12eqeltrd 2825 . 2 ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 ∈ 𝑍)
1410, 13sseldd 3974 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3941  β€˜cfv 6543  (class class class)co 7413  lecple 17234  joincjn 18297  Atomscatm 38787  HLchlt 38874  +𝑃cpadd 39320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-1st 7987  df-2nd 7988  df-padd 39321
This theorem is referenced by:  paddasslem14  39358
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