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Theorem cdlemn11pre 38232
Description: Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 38229, cdlemn11b 38230, cdlemn11c 38231, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)
Hypotheses
Ref Expression
cdlemn11a.b 𝐵 = (Base‘𝐾)
cdlemn11a.l = (le‘𝐾)
cdlemn11a.j = (join‘𝐾)
cdlemn11a.a 𝐴 = (Atoms‘𝐾)
cdlemn11a.h 𝐻 = (LHyp‘𝐾)
cdlemn11a.p 𝑃 = ((oc‘𝐾)‘𝑊)
cdlemn11a.o 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
cdlemn11a.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn11a.r 𝑅 = ((trL‘𝐾)‘𝑊)
cdlemn11a.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
cdlemn11a.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
cdlemn11a.J 𝐽 = ((DIsoC‘𝐾)‘𝑊)
cdlemn11a.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn11a.d + = (+g𝑈)
cdlemn11a.s = (LSSum‘𝑈)
cdlemn11a.f 𝐹 = (𝑇 (𝑃) = 𝑄)
cdlemn11a.g 𝐺 = (𝑇 (𝑃) = 𝑁)
Assertion
Ref Expression
cdlemn11pre (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → 𝑁 (𝑄 𝑋))
Distinct variable groups:   ,   𝐴,   𝐵,   ,𝐻   ,𝐾   ,𝑁   𝑃,   𝑄,   𝑇,   ,𝑊
Allowed substitution hints:   + ()   ()   𝑅()   𝑈()   𝐸()   𝐹()   𝐺()   𝐼()   𝐽()   ()   𝑂()   𝑋()

Proof of Theorem cdlemn11pre
Dummy variables 𝑔 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cdlemn11a.b . . 3 𝐵 = (Base‘𝐾)
2 cdlemn11a.l . . 3 = (le‘𝐾)
3 cdlemn11a.j . . 3 = (join‘𝐾)
4 cdlemn11a.a . . 3 𝐴 = (Atoms‘𝐾)
5 cdlemn11a.h . . 3 𝐻 = (LHyp‘𝐾)
6 cdlemn11a.p . . 3 𝑃 = ((oc‘𝐾)‘𝑊)
7 cdlemn11a.o . . 3 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
8 cdlemn11a.t . . 3 𝑇 = ((LTrn‘𝐾)‘𝑊)
9 cdlemn11a.r . . 3 𝑅 = ((trL‘𝐾)‘𝑊)
10 cdlemn11a.e . . 3 𝐸 = ((TEndo‘𝐾)‘𝑊)
11 cdlemn11a.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
12 cdlemn11a.J . . 3 𝐽 = ((DIsoC‘𝐾)‘𝑊)
13 cdlemn11a.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
14 cdlemn11a.d . . 3 + = (+g𝑈)
15 cdlemn11a.s . . 3 = (LSSum‘𝑈)
16 cdlemn11a.f . . 3 𝐹 = (𝑇 (𝑃) = 𝑄)
17 cdlemn11a.g . . 3 𝐺 = (𝑇 (𝑃) = 𝑁)
181, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17cdlemn11c 38231 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ∃𝑦 ∈ (𝐽𝑄)∃𝑧 ∈ (𝐼𝑋)⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧))
19 simp1 1130 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
20 simp21 1200 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
212, 4, 5, 6, 8, 10, 12, 16dicelval3 38202 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑦 ∈ (𝐽𝑄) ↔ ∃𝑠𝐸 𝑦 = ⟨(𝑠𝐹), 𝑠⟩))
2219, 20, 21syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (𝑦 ∈ (𝐽𝑄) ↔ ∃𝑠𝐸 𝑦 = ⟨(𝑠𝐹), 𝑠⟩))
23 simp23 1202 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (𝑋𝐵𝑋 𝑊))
241, 2, 5, 8, 9, 7, 11dibelval3 38169 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑧 ∈ (𝐼𝑋) ↔ ∃𝑔𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)))
2519, 23, 24syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (𝑧 ∈ (𝐼𝑋) ↔ ∃𝑔𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)))
2622, 25anbi12d 630 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ((𝑦 ∈ (𝐽𝑄) ∧ 𝑧 ∈ (𝐼𝑋)) ↔ (∃𝑠𝐸 𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ ∃𝑔𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋))))
27 reeanv 3373 . . . . 5 (∃𝑠𝐸𝑔𝑇 (𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)) ↔ (∃𝑠𝐸 𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ ∃𝑔𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)))
28 simpl1 1185 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
29 simpl21 1245 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
30 simpl22 1246 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑁𝐴 ∧ ¬ 𝑁 𝑊))
31 simpl23 1247 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑋𝐵𝑋 𝑊))
32 simpr1r 1225 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → 𝑔𝑇)
33 simpr1l 1224 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → 𝑠𝐸)
34 simpr3 1190 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))
351, 2, 4, 5, 6, 7, 8, 10, 13, 14, 16, 17cdlemn9 38227 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑔𝑄) = 𝑁)
3628, 29, 30, 33, 32, 34, 35syl123anc 1381 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑔𝑄) = 𝑁)
37 simpr2 1189 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑅𝑔) 𝑋)
381, 2, 3, 4, 5, 8, 9cdlemn10 38228 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑁 ∧ (𝑅𝑔) 𝑋)) → 𝑁 (𝑄 𝑋))
3928, 29, 30, 31, 32, 36, 37, 38syl133anc 1387 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) 𝑋 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → 𝑁 (𝑄 𝑋))
40393exp2 1348 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ((𝑠𝐸𝑔𝑇) → ((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋)))))
41 oveq12 7159 . . . . . . . . . . . . . 14 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (𝑦 + 𝑧) = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))
4241eqeq2d 2837 . . . . . . . . . . . . 13 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) ↔ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩)))
4342imbi1d 343 . . . . . . . . . . . 12 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → ((⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋)) ↔ (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋))))
4443imbi2d 342 . . . . . . . . . . 11 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋))) ↔ ((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋)))))
4544biimprd 249 . . . . . . . . . 10 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋))) → ((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋)))))
4645com23 86 . . . . . . . . 9 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → ((𝑅𝑔) 𝑋 → (((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋))) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋)))))
4746impr 455 . . . . . . . 8 ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)) → (((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋))) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋))))
4847com12 32 . . . . . . 7 (((𝑅𝑔) 𝑋 → (⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) → 𝑁 (𝑄 𝑋))) → ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋))))
4940, 48syl6 35 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ((𝑠𝐸𝑔𝑇) → ((𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋)))))
5049rexlimdvv 3298 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (∃𝑠𝐸𝑔𝑇 (𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋))))
5127, 50syl5bir 244 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ((∃𝑠𝐸 𝑦 = ⟨(𝑠𝐹), 𝑠⟩ ∧ ∃𝑔𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅𝑔) 𝑋)) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋))))
5226, 51sylbid 241 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ((𝑦 ∈ (𝐽𝑄) ∧ 𝑧 ∈ (𝐼𝑋)) → (⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋))))
5352rexlimdvv 3298 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → (∃𝑦 ∈ (𝐽𝑄)∃𝑧 ∈ (𝐼𝑋)⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧) → 𝑁 (𝑄 𝑋)))
5418, 53mpd 15 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → 𝑁 (𝑄 𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2107  wrex 3144  wss 3940  cop 4570   class class class wbr 5063  cmpt 5143   I cid 5458  cres 5556  cfv 6354  crio 7107  (class class class)co 7150  Basecbs 16478  +gcplusg 16560  lecple 16567  occoc 16568  joincjn 17549  LSSumclsm 18695  Atomscatm 36285  HLchlt 36372  LHypclh 37006  LTrncltrn 37123  trLctrl 37180  TEndoctendo 37774  DVecHcdvh 38100  DIsoBcdib 38160  DIsoCcdic 38194
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-riotaBAD 35975
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-tpos 7888  df-undef 7935  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-0g 16710  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18051  df-minusg 18052  df-sbg 18053  df-subg 18221  df-lsm 18697  df-mgp 19176  df-ur 19188  df-ring 19235  df-oppr 19309  df-dvdsr 19327  df-unit 19328  df-invr 19358  df-dvr 19369  df-drng 19440  df-lmod 19572  df-lss 19640  df-lvec 19811  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-llines 36520  df-lplanes 36521  df-lvols 36522  df-lines 36523  df-psubsp 36525  df-pmap 36526  df-padd 36818  df-lhyp 37010  df-laut 37011  df-ldil 37126  df-ltrn 37127  df-trl 37181  df-tendo 37777  df-edring 37779  df-disoa 38051  df-dvech 38101  df-dib 38161  df-dic 38195
This theorem is referenced by:  cdlemn11  38233
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