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Theorem cdleme50trn3 40078
Description: Part of proof that 𝐹 is a translation. 𝑃 = 𝑄 case. TODO: fix comment. (Contributed by NM, 10-Apr-2013.)
Hypotheses
Ref Expression
cdlemef50.b 𝐡 = (Baseβ€˜πΎ)
cdlemef50.l ≀ = (leβ€˜πΎ)
cdlemef50.j ∨ = (joinβ€˜πΎ)
cdlemef50.m ∧ = (meetβ€˜πΎ)
cdlemef50.a 𝐴 = (Atomsβ€˜πΎ)
cdlemef50.h 𝐻 = (LHypβ€˜πΎ)
cdlemef50.u π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
cdlemef50.d 𝐷 = ((𝑑 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑑) ∧ π‘Š)))
cdlemefs50.e 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑑) ∧ π‘Š)))
cdlemef50.f 𝐹 = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (π‘₯ ∧ π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑦 = 𝐸)), ⦋𝑠 / π‘‘β¦Œπ·) ∨ (π‘₯ ∧ π‘Š)))), π‘₯))
Assertion
Ref Expression
cdleme50trn3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ ((𝑅 ∨ (πΉβ€˜π‘…)) ∧ π‘Š) = π‘ˆ)
Distinct variable groups:   𝑑,𝑠,π‘₯,𝑦,𝑧, ∧   ∨ ,𝑠,𝑑,π‘₯,𝑦,𝑧   ≀ ,𝑠,𝑑,π‘₯,𝑦,𝑧   𝐴,𝑠,𝑑,π‘₯,𝑦,𝑧   𝐡,𝑠,𝑑,π‘₯,𝑦,𝑧   𝐷,𝑠,π‘₯,𝑦,𝑧   π‘₯,𝐸,𝑦,𝑧   𝐻,𝑠,𝑑,π‘₯,𝑦,𝑧   𝐾,𝑠,𝑑,π‘₯,𝑦,𝑧   𝑃,𝑠,𝑑,π‘₯,𝑦,𝑧   𝑄,𝑠,𝑑,π‘₯,𝑦,𝑧   𝑅,𝑠,𝑑,π‘₯,𝑦,𝑧   π‘ˆ,𝑠,𝑑,π‘₯,𝑦,𝑧   π‘Š,𝑠,𝑑,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐷(𝑑)   𝐸(𝑑,𝑠)   𝐹(π‘₯,𝑦,𝑧,𝑑,𝑠)

Proof of Theorem cdleme50trn3
StepHypRef Expression
1 simpl1 1188 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 simprr 771 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))
3 cdlemef50.l . . . . . 6 ≀ = (leβ€˜πΎ)
4 cdlemef50.m . . . . . 6 ∧ = (meetβ€˜πΎ)
5 eqid 2725 . . . . . 6 (0.β€˜πΎ) = (0.β€˜πΎ)
6 cdlemef50.a . . . . . 6 𝐴 = (Atomsβ€˜πΎ)
7 cdlemef50.h . . . . . 6 𝐻 = (LHypβ€˜πΎ)
83, 4, 5, 6, 7lhpmat 39555 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ (𝑅 ∧ π‘Š) = (0.β€˜πΎ))
91, 2, 8syl2anc 582 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑅 ∧ π‘Š) = (0.β€˜πΎ))
10 simprrl 779 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝑅 ∈ 𝐴)
11 cdlemef50.b . . . . . . . . . 10 𝐡 = (Baseβ€˜πΎ)
1211, 6atbase 38813 . . . . . . . . 9 (𝑅 ∈ 𝐴 β†’ 𝑅 ∈ 𝐡)
1310, 12syl 17 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝑅 ∈ 𝐡)
14 simprl 769 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝑃 = 𝑄)
15 cdlemef50.f . . . . . . . . 9 𝐹 = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠 ∨ (π‘₯ ∧ π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑦 = 𝐸)), ⦋𝑠 / π‘‘β¦Œπ·) ∨ (π‘₯ ∧ π‘Š)))), π‘₯))
1615cdleme31id 39919 . . . . . . . 8 ((𝑅 ∈ 𝐡 ∧ 𝑃 = 𝑄) β†’ (πΉβ€˜π‘…) = 𝑅)
1713, 14, 16syl2anc 582 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (πΉβ€˜π‘…) = 𝑅)
1817oveq2d 7429 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑅 ∨ (πΉβ€˜π‘…)) = (𝑅 ∨ 𝑅))
19 simpl1l 1221 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝐾 ∈ HL)
20 cdlemef50.j . . . . . . . 8 ∨ = (joinβ€˜πΎ)
2120, 6hlatjidm 38893 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑅 ∈ 𝐴) β†’ (𝑅 ∨ 𝑅) = 𝑅)
2219, 10, 21syl2anc 582 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑅 ∨ 𝑅) = 𝑅)
2318, 22eqtrd 2765 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑅 ∨ (πΉβ€˜π‘…)) = 𝑅)
2423oveq1d 7428 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ ((𝑅 ∨ (πΉβ€˜π‘…)) ∧ π‘Š) = (𝑅 ∧ π‘Š))
25 simpl2 1189 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
263, 4, 5, 6, 7lhpmat 39555 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃 ∧ π‘Š) = (0.β€˜πΎ))
271, 25, 26syl2anc 582 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑃 ∧ π‘Š) = (0.β€˜πΎ))
289, 24, 273eqtr4d 2775 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ ((𝑅 ∨ (πΉβ€˜π‘…)) ∧ π‘Š) = (𝑃 ∧ π‘Š))
29 simpl2l 1223 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝑃 ∈ 𝐴)
3020, 6hlatjidm 38893 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) β†’ (𝑃 ∨ 𝑃) = 𝑃)
3119, 29, 30syl2anc 582 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑃 ∨ 𝑃) = 𝑃)
3214oveq2d 7429 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑃 ∨ 𝑃) = (𝑃 ∨ 𝑄))
3331, 32eqtr3d 2767 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝑃 = (𝑃 ∨ 𝑄))
3433oveq1d 7428 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑃 ∧ π‘Š) = ((𝑃 ∨ 𝑄) ∧ π‘Š))
3528, 34eqtrd 2765 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ ((𝑅 ∨ (πΉβ€˜π‘…)) ∧ π‘Š) = ((𝑃 ∨ 𝑄) ∧ π‘Š))
36 cdlemef50.u . 2 π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)
3735, 36eqtr4di 2783 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 = 𝑄 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ ((𝑅 ∨ (πΉβ€˜π‘…)) ∧ π‘Š) = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  β¦‹csb 3886  ifcif 4525   class class class wbr 5144   ↦ cmpt 5227  β€˜cfv 6543  β„©crio 7368  (class class class)co 7413  Basecbs 17174  lecple 17234  joincjn 18297  meetcmee 18298  0.cp0 18409  Atomscatm 38787  HLchlt 38874  LHypclh 39509
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-rmo 3364  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-riota 7369  df-ov 7416  df-oprab 7417  df-proset 18281  df-poset 18299  df-plt 18316  df-lub 18332  df-glb 18333  df-join 18334  df-meet 18335  df-p0 18411  df-lat 18418  df-covers 38790  df-ats 38791  df-atl 38822  df-cvlat 38846  df-hlat 38875  df-lhyp 39513
This theorem is referenced by:  cdleme50trn123  40079
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