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Theorem cdlemg2idN 39771
Description: Version of cdleme31id 39569 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 21-Apr-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
cdlemg2id.l ≀ = (leβ€˜πΎ)
cdlemg2id.a 𝐴 = (Atomsβ€˜πΎ)
cdlemg2id.h 𝐻 = (LHypβ€˜πΎ)
cdlemg2id.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
cdlemg2id.b 𝐡 = (Baseβ€˜πΎ)
Assertion
Ref Expression
cdlemg2idN ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ (πΉβ€˜π‘‹) = 𝑋)
Proof of Theorem cdlemg2idN
Dummy variables 𝑑 𝑠 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp111 1301 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ 𝐾 ∈ HL)
2 simp112 1302 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ π‘Š ∈ 𝐻)
3 simp12 1203 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))
4 simp13 1204 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))
5 simp113 1303 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ 𝐹 ∈ 𝑇)
6 simp2l 1198 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ (πΉβ€˜π‘ƒ) = 𝑄)
7 cdlemg2id.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
8 cdlemg2id.l . . . . 5 ≀ = (leβ€˜πΎ)
9 eqid 2731 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
10 eqid 2731 . . . . 5 (meetβ€˜πΎ) = (meetβ€˜πΎ)
11 cdlemg2id.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
12 cdlemg2id.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
13 cdlemg2id.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
14 eqid 2731 . . . . 5 ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š) = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š)
15 eqid 2731 . . . . 5 ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))) = ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))
16 eqid 2731 . . . . 5 ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))) = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))
17 eqid 2731 . . . . 5 (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯)) = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯))
187, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemg2dN 39765 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑄)) β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯)))
191, 2, 3, 4, 5, 6, 18syl222anc 1385 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯)))
2019fveq1d 6894 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ (πΉβ€˜π‘‹) = ((π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯))β€˜π‘‹))
21 simp2r 1199 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ 𝑋 ∈ 𝐡)
22 simp3 1137 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ 𝑃 = 𝑄)
2317cdleme31id 39569 . . 3 ((𝑋 ∈ 𝐡 ∧ 𝑃 = 𝑄) β†’ ((π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯))β€˜π‘‹) = 𝑋)
2421, 22, 23syl2anc 583 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ ((π‘₯ ∈ 𝐡 ↦ if((𝑃 β‰  𝑄 ∧ Β¬ π‘₯ ≀ π‘Š), (℩𝑧 ∈ 𝐡 βˆ€π‘  ∈ 𝐴 ((Β¬ 𝑠 ≀ π‘Š ∧ (𝑠(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)) = π‘₯) β†’ 𝑧 = (if(𝑠 ≀ (𝑃(joinβ€˜πΎ)𝑄), (℩𝑦 ∈ 𝐡 βˆ€π‘‘ ∈ 𝐴 ((Β¬ 𝑑 ≀ π‘Š ∧ Β¬ 𝑑 ≀ (𝑃(joinβ€˜πΎ)𝑄)) β†’ 𝑦 = ((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)(((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š)))(joinβ€˜πΎ)((𝑠(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))), ⦋𝑠 / π‘‘β¦Œ((𝑑(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑄)(meetβ€˜πΎ)π‘Š))(meetβ€˜πΎ)(𝑄(joinβ€˜πΎ)((𝑃(joinβ€˜πΎ)𝑑)(meetβ€˜πΎ)π‘Š))))(joinβ€˜πΎ)(π‘₯(meetβ€˜πΎ)π‘Š)))), π‘₯))β€˜π‘‹) = 𝑋)
2520, 24eqtrd 2771 1 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = 𝑄 ∧ 𝑋 ∈ 𝐡) ∧ 𝑃 = 𝑄) β†’ (πΉβ€˜π‘‹) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  β¦‹csb 3894  ifcif 4529   class class class wbr 5149   ↦ cmpt 5232  β€˜cfv 6544  β„©crio 7367  (class class class)co 7412  Basecbs 17149  lecple 17209  joincjn 18269  meetcmee 18270  Atomscatm 38437  HLchlt 38524  LHypclh 39159  LTrncltrn 39276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-riotaBAD 38127
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-undef 8261  df-map 8825  df-proset 18253  df-poset 18271  df-plt 18288  df-lub 18304  df-glb 18305  df-join 18306  df-meet 18307  df-p0 18383  df-p1 18384  df-lat 18390  df-clat 18457  df-oposet 38350  df-ol 38352  df-oml 38353  df-covers 38440  df-ats 38441  df-atl 38472  df-cvlat 38496  df-hlat 38525  df-llines 38673  df-lplanes 38674  df-lvols 38675  df-lines 38676  df-psubsp 38678  df-pmap 38679  df-padd 38971  df-lhyp 39163  df-laut 39164  df-ldil 39279  df-ltrn 39280  df-trl 39334
This theorem is referenced by: (None)
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