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Theorem dihopelvalcpre 38266
Description: Member of value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊, given auxiliary atom 𝑄. TODO: refactor to be shorter and more understandable; add lemmas? (Contributed by NM, 13-Mar-2014.)
Hypotheses
Ref Expression
dihopelvalcp.b 𝐵 = (Base‘𝐾)
dihopelvalcp.l = (le‘𝐾)
dihopelvalcp.j = (join‘𝐾)
dihopelvalcp.m = (meet‘𝐾)
dihopelvalcp.a 𝐴 = (Atoms‘𝐾)
dihopelvalcp.h 𝐻 = (LHyp‘𝐾)
dihopelvalcp.p 𝑃 = ((oc‘𝐾)‘𝑊)
dihopelvalcp.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihopelvalcp.r 𝑅 = ((trL‘𝐾)‘𝑊)
dihopelvalcp.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihopelvalcp.i 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihopelvalcp.g 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
dihopelvalcp.f 𝐹 ∈ V
dihopelvalcp.s 𝑆 ∈ V
dihopelvalcp.z 𝑍 = (𝑇 ↦ ( I ↾ 𝐵))
dihopelvalcp.n 𝑁 = ((DIsoB‘𝐾)‘𝑊)
dihopelvalcp.c 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihopelvalcp.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihopelvalcp.d + = (+g𝑈)
dihopelvalcp.v 𝑉 = (LSubSp‘𝑈)
dihopelvalcp.y = (LSSum‘𝑈)
dihopelvalcp.o 𝑂 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑇 ↦ ((𝑎) ∘ (𝑏))))
Assertion
Ref Expression
dihopelvalcpre (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
Distinct variable groups:   ,𝑔   𝐴,𝑔   𝑃,𝑔   𝑎,𝑏,𝐸   𝑔,,𝐻   𝑔,𝑎,,𝐾,𝑏   𝐵,   𝑇,𝑎,𝑏,𝑔,   𝑊,𝑎,𝑏,𝑔,   𝑄,𝑔
Allowed substitution hints:   𝐴(,𝑎,𝑏)   𝐵(𝑔,𝑎,𝑏)   𝐶(𝑔,,𝑎,𝑏)   𝑃(,𝑎,𝑏)   + (𝑔,,𝑎,𝑏)   (𝑔,,𝑎,𝑏)   𝑄(,𝑎,𝑏)   𝑅(𝑔,,𝑎,𝑏)   𝑆(𝑔,,𝑎,𝑏)   𝑈(𝑔,,𝑎,𝑏)   𝐸(𝑔,)   𝐹(𝑔,,𝑎,𝑏)   𝐺(𝑔,,𝑎,𝑏)   𝐻(𝑎,𝑏)   𝐼(𝑔,,𝑎,𝑏)   (𝑔,,𝑎,𝑏)   (,𝑎,𝑏)   (𝑔,,𝑎,𝑏)   𝑁(𝑔,,𝑎,𝑏)   𝑂(𝑔,,𝑎,𝑏)   𝑉(𝑔,,𝑎,𝑏)   𝑋(𝑔,,𝑎,𝑏)   𝑍(𝑔,,𝑎,𝑏)

Proof of Theorem dihopelvalcpre
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dihopelvalcp.b . . . 4 𝐵 = (Base‘𝐾)
2 dihopelvalcp.l . . . 4 = (le‘𝐾)
3 dihopelvalcp.j . . . 4 = (join‘𝐾)
4 dihopelvalcp.m . . . 4 = (meet‘𝐾)
5 dihopelvalcp.a . . . 4 𝐴 = (Atoms‘𝐾)
6 dihopelvalcp.h . . . 4 𝐻 = (LHyp‘𝐾)
7 dihopelvalcp.i . . . 4 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8 dihopelvalcp.n . . . 4 𝑁 = ((DIsoB‘𝐾)‘𝑊)
9 dihopelvalcp.c . . . 4 𝐶 = ((DIsoC‘𝐾)‘𝑊)
10 dihopelvalcp.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
11 dihopelvalcp.y . . . 4 = (LSSum‘𝑈)
121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11dihvalcq 38254 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝑁‘(𝑋 𝑊))))
1312eleq2d 2898 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐶𝑄) (𝑁‘(𝑋 𝑊)))))
14 simp1 1128 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 simp3l 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
16 dihopelvalcp.v . . . . 5 𝑉 = (LSubSp‘𝑈)
172, 5, 6, 10, 9, 16diclss 38211 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐶𝑄) ∈ 𝑉)
1814, 15, 17syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐶𝑄) ∈ 𝑉)
19 simp1l 1189 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
2019hllatd 36382 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21 simp2l 1191 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
22 simp1r 1190 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
231, 6lhpbase 37016 . . . . . 6 (𝑊𝐻𝑊𝐵)
2422, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
251, 4latmcl 17652 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
2620, 21, 24, 25syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) ∈ 𝐵)
271, 2, 4latmle2 17677 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
2820, 21, 24, 27syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) 𝑊)
291, 2, 6, 10, 8, 16diblss 38188 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (𝑁‘(𝑋 𝑊)) ∈ 𝑉)
3014, 26, 28, 29syl12anc 832 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑁‘(𝑋 𝑊)) ∈ 𝑉)
31 dihopelvalcp.d . . . 4 + = (+g𝑈)
326, 10, 31, 16, 11dvhopellsm 38135 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐶𝑄) ∈ 𝑉 ∧ (𝑁‘(𝑋 𝑊)) ∈ 𝑉) → (⟨𝐹, 𝑆⟩ ∈ ((𝐶𝑄) (𝑁‘(𝑋 𝑊))) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩))))
3314, 18, 30, 32syl3anc 1363 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ ((𝐶𝑄) (𝑁‘(𝑋 𝑊))) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩))))
34 dihopelvalcp.p . . . . . . . . 9 𝑃 = ((oc‘𝐾)‘𝑊)
35 dihopelvalcp.t . . . . . . . . 9 𝑇 = ((LTrn‘𝐾)‘𝑊)
36 dihopelvalcp.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
37 dihopelvalcp.g . . . . . . . . 9 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
38 vex 3498 . . . . . . . . 9 𝑥 ∈ V
39 vex 3498 . . . . . . . . 9 𝑦 ∈ V
402, 5, 6, 34, 35, 36, 9, 37, 38, 39dicopelval2 38199 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ↔ (𝑥 = (𝑦𝐺) ∧ 𝑦𝐸)))
4114, 15, 40syl2anc 584 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ↔ (𝑥 = (𝑦𝐺) ∧ 𝑦𝐸)))
42 dihopelvalcp.r . . . . . . . . 9 𝑅 = ((trL‘𝐾)‘𝑊)
43 dihopelvalcp.z . . . . . . . . 9 𝑍 = (𝑇 ↦ ( I ↾ 𝐵))
441, 2, 6, 35, 42, 43, 8dibopelval3 38166 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊)) ↔ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)))
4514, 26, 28, 44syl12anc 832 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊)) ↔ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)))
4641, 45anbi12d 630 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ↔ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))))
4746anbi1d 629 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩)) ↔ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩))))
48 simpl1 1183 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
49 simprll 775 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑥 = (𝑦𝐺))
50 simprlr 776 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑦𝐸)
512, 5, 6, 34lhpocnel2 37037 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
5248, 51syl 17 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
53 simpl3l 1220 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
542, 5, 6, 35, 37ltrniotacl 37597 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
5548, 52, 53, 54syl3anc 1363 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝐺𝑇)
566, 35, 36tendocl 37785 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑦𝐸𝐺𝑇) → (𝑦𝐺) ∈ 𝑇)
5748, 50, 55, 56syl3anc 1363 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝐺) ∈ 𝑇)
5849, 57eqeltrd 2913 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑥𝑇)
59 simprll 775 . . . . . . . . . . . 12 (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) → 𝑧𝑇)
6059adantl 482 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑧𝑇)
61 simprrr 778 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑤 = 𝑍)
621, 6, 35, 36, 43tendo0cl 37808 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑍𝐸)
6348, 62syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑍𝐸)
6461, 63eqeltrd 2913 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑤𝐸)
65 eqid 2821 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
66 eqid 2821 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
676, 35, 36, 10, 65, 31, 66dvhopvadd 38111 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑦𝐸) ∧ (𝑧𝑇𝑤𝐸)) → (⟨𝑥, 𝑦+𝑧, 𝑤⟩) = ⟨(𝑥𝑧), (𝑦(+g‘(Scalar‘𝑈))𝑤)⟩)
6848, 58, 50, 60, 64, 67syl122anc 1371 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝑥, 𝑦+𝑧, 𝑤⟩) = ⟨(𝑥𝑧), (𝑦(+g‘(Scalar‘𝑈))𝑤)⟩)
69 dihopelvalcp.o . . . . . . . . . . . . . 14 𝑂 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑇 ↦ ((𝑎) ∘ (𝑏))))
706, 35, 36, 10, 65, 69, 66dvhfplusr 38102 . . . . . . . . . . . . 13 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = 𝑂)
7148, 70syl 17 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (+g‘(Scalar‘𝑈)) = 𝑂)
7271oveqd 7162 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦(+g‘(Scalar‘𝑈))𝑤) = (𝑦𝑂𝑤))
7372opeq2d 4804 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → ⟨(𝑥𝑧), (𝑦(+g‘(Scalar‘𝑈))𝑤)⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩)
7468, 73eqtrd 2856 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝑥, 𝑦+𝑧, 𝑤⟩) = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩)
7574eqeq2d 2832 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩))
76 dihopelvalcp.f . . . . . . . . . 10 𝐹 ∈ V
77 dihopelvalcp.s . . . . . . . . . 10 𝑆 ∈ V
7876, 77opth 5360 . . . . . . . . 9 (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
7961oveq2d 7161 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝑂𝑤) = (𝑦𝑂𝑍))
801, 6, 35, 36, 43, 69tendo0plr 37810 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑦𝐸) → (𝑦𝑂𝑍) = 𝑦)
8148, 50, 80syl2anc 584 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝑂𝑍) = 𝑦)
8279, 81eqtrd 2856 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝑂𝑤) = 𝑦)
8382eqeq2d 2832 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑆 = (𝑦𝑂𝑤) ↔ 𝑆 = 𝑦))
8483anbi2d 628 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → ((𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)) ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)))
8578, 84syl5bb 284 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)))
8675, 85bitrd 280 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩) ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)))
8786pm5.32da 579 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩)) ↔ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))))
88 simplll 771 . . . . . . . . . . 11 ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → 𝑥 = (𝑦𝐺))
8988adantl 482 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 = (𝑦𝐺))
90 simprrr 778 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑆 = 𝑦)
9190fveq1d 6666 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑆𝐺) = (𝑦𝐺))
9289, 91eqtr4d 2859 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 = (𝑆𝐺))
9390eqcomd 2827 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑦 = 𝑆)
94 coass 6112 . . . . . . . . . . 11 (((𝑆𝐺) ∘ (𝑆𝐺)) ∘ 𝑧) = ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧))
95 simpl1 1183 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
96 simpllr 772 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → 𝑦𝐸)
9796adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑦𝐸)
9890, 97eqeltrd 2913 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑆𝐸)
9955adantrr 713 . . . . . . . . . . . . . . . 16 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐺𝑇)
1006, 35, 36tendocl 37785 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝐺𝑇) → (𝑆𝐺) ∈ 𝑇)
10195, 98, 99, 100syl3anc 1363 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑆𝐺) ∈ 𝑇)
1021, 6, 35ltrn1o 37142 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇) → (𝑆𝐺):𝐵1-1-onto𝐵)
10395, 101, 102syl2anc 584 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑆𝐺):𝐵1-1-onto𝐵)
104 f1ococnv1 6637 . . . . . . . . . . . . . 14 ((𝑆𝐺):𝐵1-1-onto𝐵 → ((𝑆𝐺) ∘ (𝑆𝐺)) = ( I ↾ 𝐵))
105103, 104syl 17 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → ((𝑆𝐺) ∘ (𝑆𝐺)) = ( I ↾ 𝐵))
106105coeq1d 5726 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (((𝑆𝐺) ∘ (𝑆𝐺)) ∘ 𝑧) = (( I ↾ 𝐵) ∘ 𝑧))
10759ad2antrl 724 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑧𝑇)
1081, 6, 35ltrn1o 37142 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑧𝑇) → 𝑧:𝐵1-1-onto𝐵)
10995, 107, 108syl2anc 584 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑧:𝐵1-1-onto𝐵)
110 f1of 6609 . . . . . . . . . . . . 13 (𝑧:𝐵1-1-onto𝐵𝑧:𝐵𝐵)
111 fcoi2 6547 . . . . . . . . . . . . 13 (𝑧:𝐵𝐵 → (( I ↾ 𝐵) ∘ 𝑧) = 𝑧)
112109, 110, 1113syl 18 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (( I ↾ 𝐵) ∘ 𝑧) = 𝑧)
113106, 112eqtr2d 2857 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑧 = (((𝑆𝐺) ∘ (𝑆𝐺)) ∘ 𝑧))
114 simprrl 777 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 = (𝑥𝑧))
11592coeq1d 5726 . . . . . . . . . . . . . 14 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑥𝑧) = ((𝑆𝐺) ∘ 𝑧))
116114, 115eqtrd 2856 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 = ((𝑆𝐺) ∘ 𝑧))
117116coeq1d 5726 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝐹(𝑆𝐺)) = (((𝑆𝐺) ∘ 𝑧) ∘ (𝑆𝐺)))
1186, 35ltrncnv 37164 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇) → (𝑆𝐺) ∈ 𝑇)
11995, 101, 118syl2anc 584 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑆𝐺) ∈ 𝑇)
1206, 35ltrnco 37737 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇𝑧𝑇) → ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇)
12195, 101, 107, 120syl3anc 1363 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇)
1226, 35ltrncom 37756 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇) → ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧)) = (((𝑆𝐺) ∘ 𝑧) ∘ (𝑆𝐺)))
12395, 119, 121, 122syl3anc 1363 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧)) = (((𝑆𝐺) ∘ 𝑧) ∘ (𝑆𝐺)))
124117, 123eqtr4d 2859 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝐹(𝑆𝐺)) = ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧)))
12594, 113, 1243eqtr4a 2882 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑧 = (𝐹(𝑆𝐺)))
126 simplrr 774 . . . . . . . . . . 11 ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → 𝑤 = 𝑍)
127126adantl 482 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑤 = 𝑍)
128125, 127jca 512 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))
12992, 93, 128jca31 515 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)))
130129ex 413 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))))
131130pm4.71rd 563 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) ↔ (((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)))))
13287, 131bitrd 280 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩)) ↔ (((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)))))
133 simprrl 777 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 = (𝑥𝑧))
134 simpll1 1204 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
13588adantl 482 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 = (𝑦𝐺))
13696adantl 482 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑦𝐸)
137134, 51syl 17 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
138 simpl3l 1220 . . . . . . . . . . . . . . 15 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
139138adantr 481 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
140134, 137, 139, 54syl3anc 1363 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐺𝑇)
141134, 136, 140, 56syl3anc 1363 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑦𝐺) ∈ 𝑇)
142135, 141eqeltrd 2913 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑥𝑇)
14359ad2antrl 724 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑧𝑇)
1446, 35ltrnco 37737 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥𝑇𝑧𝑇) → (𝑥𝑧) ∈ 𝑇)
145134, 142, 143, 144syl3anc 1363 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑥𝑧) ∈ 𝑇)
146133, 145eqeltrd 2913 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐹𝑇)
147 simpl1l 1216 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝐾 ∈ HL)
148147adantr 481 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐾 ∈ HL)
149148hllatd 36382 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝐾 ∈ Lat)
1501, 6, 35, 42trlcl 37182 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑧𝑇) → (𝑅𝑧) ∈ 𝐵)
151134, 143, 150syl2anc 584 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑅𝑧) ∈ 𝐵)
152 simpl2l 1218 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝑋𝐵)
153152adantr 481 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑋𝐵)
154 simpl1r 1217 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝑊𝐻)
155154adantr 481 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑊𝐻)
156155, 23syl 17 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → 𝑊𝐵)
157149, 153, 156, 25syl3anc 1363 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑋 𝑊) ∈ 𝐵)
158 simprlr 776 . . . . . . . . . . 11 (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅𝑧) (𝑋 𝑊))
159158ad2antrl 724 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑅𝑧) (𝑋 𝑊))
1601, 2, 4latmle1 17676 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑋)
161149, 153, 156, 160syl3anc 1363 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑋 𝑊) 𝑋)
1621, 2, 149, 151, 157, 153, 159, 161lattrd 17658 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → (𝑅𝑧) 𝑋)
163146, 136, 162jca31 515 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) → ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋))
164 simprll 775 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝑥 = (𝑆𝐺))
165164adantr 481 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑥 = (𝑆𝐺))
166 simprlr 776 . . . . . . . . . . . . 13 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝑦 = 𝑆)
167166adantr 481 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑦 = 𝑆)
168167fveq1d 6666 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑦𝐺) = (𝑆𝐺))
169165, 168eqtr4d 2859 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑥 = (𝑦𝐺))
170 simprlr 776 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑦𝐸)
171169, 170jca 512 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑥 = (𝑦𝐺) ∧ 𝑦𝐸))
172 simprrl 777 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝑧 = (𝐹(𝑆𝐺)))
173172adantr 481 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑧 = (𝐹(𝑆𝐺)))
174 simpll1 1204 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
175 simprll 775 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐹𝑇)
176167, 170eqeltrrd 2914 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑆𝐸)
177174, 51syl 17 . . . . . . . . . . . . . . 15 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))
178138adantr 481 . . . . . . . . . . . . . . 15 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
179174, 177, 178, 54syl3anc 1363 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐺𝑇)
180174, 176, 179, 100syl3anc 1363 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑆𝐺) ∈ 𝑇)
181174, 180, 118syl2anc 584 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑆𝐺) ∈ 𝑇)
1826, 35ltrnco 37737 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇(𝑆𝐺) ∈ 𝑇) → (𝐹(𝑆𝐺)) ∈ 𝑇)
183174, 175, 181, 182syl3anc 1363 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝐹(𝑆𝐺)) ∈ 𝑇)
184173, 183eqeltrd 2913 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑧𝑇)
185 simprr 769 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑅𝑧) 𝑋)
1862, 6, 35, 42trlle 37202 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑧𝑇) → (𝑅𝑧) 𝑊)
187174, 184, 186syl2anc 584 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑅𝑧) 𝑊)
188147adantr 481 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐾 ∈ HL)
189188hllatd 36382 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐾 ∈ Lat)
190174, 184, 150syl2anc 584 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑅𝑧) ∈ 𝐵)
191152adantr 481 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑋𝐵)
192154adantr 481 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑊𝐻)
193192, 23syl 17 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑊𝐵)
1941, 2, 4latlem12 17678 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ ((𝑅𝑧) ∈ 𝐵𝑋𝐵𝑊𝐵)) → (((𝑅𝑧) 𝑋 ∧ (𝑅𝑧) 𝑊) ↔ (𝑅𝑧) (𝑋 𝑊)))
195189, 190, 191, 193, 194syl13anc 1364 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (((𝑅𝑧) 𝑋 ∧ (𝑅𝑧) 𝑊) ↔ (𝑅𝑧) (𝑋 𝑊)))
196185, 187, 195mpbi2and 708 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑅𝑧) (𝑋 𝑊))
197 simprrr 778 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → 𝑤 = 𝑍)
198197adantr 481 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑤 = 𝑍)
199184, 196, 198jca31 515 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍))
200174, 180, 102syl2anc 584 . . . . . . . . . . . . . . . 16 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑆𝐺):𝐵1-1-onto𝐵)
201200, 104syl 17 . . . . . . . . . . . . . . 15 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → ((𝑆𝐺) ∘ (𝑆𝐺)) = ( I ↾ 𝐵))
202201coeq2d 5727 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺))) = (𝐹 ∘ ( I ↾ 𝐵)))
2031, 6, 35ltrn1o 37142 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
204174, 175, 203syl2anc 584 . . . . . . . . . . . . . . 15 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐹:𝐵1-1-onto𝐵)
205 f1of 6609 . . . . . . . . . . . . . . 15 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
206 fcoi1 6546 . . . . . . . . . . . . . . 15 (𝐹:𝐵𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
207204, 205, 2063syl 18 . . . . . . . . . . . . . 14 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
208202, 207eqtr2d 2857 . . . . . . . . . . . . 13 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐹 = (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺))))
209 coass 6112 . . . . . . . . . . . . 13 ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)) = (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺)))
210208, 209syl6eqr 2874 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐹 = ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)))
2116, 35ltrncom 37756 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ (𝐹(𝑆𝐺)) ∈ 𝑇) → ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))) = ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)))
212174, 180, 183, 211syl3anc 1363 . . . . . . . . . . . 12 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))) = ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)))
213210, 212eqtr4d 2859 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐹 = ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))))
214165, 173coeq12d 5729 . . . . . . . . . . 11 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝑥𝑧) = ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))))
215213, 214eqtr4d 2859 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝐹 = (𝑥𝑧))
216167eqcomd 2827 . . . . . . . . . 10 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → 𝑆 = 𝑦)
217215, 216jca 512 . . . . . . . . 9 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))
218171, 199, 217jca31 515 . . . . . . . 8 (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) → (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)))
219163, 218impbida 797 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍))) → ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) ↔ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
220219pm5.32da 579 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) ↔ (((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋))))
221 df-3an 1081 . . . . . 6 (((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) ↔ (((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
222220, 221syl6bbr 290 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → ((((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦))) ↔ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋))))
22347, 132, 2223bitrd 306 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩)) ↔ ((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋))))
2242234exbidv 1918 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩)) ↔ ∃𝑥𝑦𝑧𝑤((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋))))
225 fvex 6677 . . . 4 (𝑆𝐺) ∈ V
226225cnvex 7618 . . . . 5 (𝑆𝐺) ∈ V
22776, 226coex 7623 . . . 4 (𝐹(𝑆𝐺)) ∈ V
22835fvexi 6678 . . . . . 6 𝑇 ∈ V
229228mptex 6978 . . . . 5 (𝑇 ↦ ( I ↾ 𝐵)) ∈ V
23043, 229eqeltri 2909 . . . 4 𝑍 ∈ V
231 biidd 263 . . . 4 (𝑥 = (𝑆𝐺) → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
232 eleq1 2900 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝐸𝑆𝐸))
233232anbi2d 628 . . . . 5 (𝑦 = 𝑆 → ((𝐹𝑇𝑦𝐸) ↔ (𝐹𝑇𝑆𝐸)))
234233anbi1d 629 . . . 4 (𝑦 = 𝑆 → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋)))
235 fveq2 6664 . . . . . 6 (𝑧 = (𝐹(𝑆𝐺)) → (𝑅𝑧) = (𝑅‘(𝐹(𝑆𝐺))))
236235breq1d 5068 . . . . 5 (𝑧 = (𝐹(𝑆𝐺)) → ((𝑅𝑧) 𝑋 ↔ (𝑅‘(𝐹(𝑆𝐺))) 𝑋))
237236anbi2d 628 . . . 4 (𝑧 = (𝐹(𝑆𝐺)) → (((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
238 biidd 263 . . . 4 (𝑤 = 𝑍 → (((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
239225, 77, 227, 230, 231, 234, 237, 238ceqsex4v 3547 . . 3 (∃𝑥𝑦𝑧𝑤((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋))
240224, 239syl6bb 288 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩)) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
24113, 33, 2403bitrd 306 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  Vcvv 3495  cop 4565   class class class wbr 5058  cmpt 5138   I cid 5453  ccnv 5548  cres 5551  ccom 5553  wf 6345  1-1-ontowf1o 6348  cfv 6349  crio 7102  (class class class)co 7145  cmpo 7147  Basecbs 16473  +gcplusg 16555  Scalarcsca 16558  lecple 16562  occoc 16563  joincjn 17544  meetcmee 17545  Latclat 17645  LSSumclsm 18690  LSubSpclss 19634  Atomscatm 36281  HLchlt 36368  LHypclh 37002  LTrncltrn 37119  trLctrl 37176  TEndoctendo 37770  DVecHcdvh 38096  DIsoBcdib 38156  DIsoCcdic 38190  DIsoHcdih 38246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-riotaBAD 35971
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-tpos 7883  df-undef 7930  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-n0 11887  df-z 11971  df-uz 12233  df-fz 12883  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-mulr 16569  df-sca 16571  df-vsca 16572  df-0g 16705  df-proset 17528  df-poset 17546  df-plt 17558  df-lub 17574  df-glb 17575  df-join 17576  df-meet 17577  df-p0 17639  df-p1 17640  df-lat 17646  df-clat 17708  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-submnd 17947  df-grp 18046  df-minusg 18047  df-sbg 18048  df-subg 18216  df-cntz 18387  df-lsm 18692  df-cmn 18839  df-abl 18840  df-mgp 19171  df-ur 19183  df-ring 19230  df-oppr 19304  df-dvdsr 19322  df-unit 19323  df-invr 19353  df-dvr 19364  df-drng 19435  df-lmod 19567  df-lss 19635  df-lsp 19675  df-lvec 19806  df-oposet 36194  df-ol 36196  df-oml 36197  df-covers 36284  df-ats 36285  df-atl 36316  df-cvlat 36340  df-hlat 36369  df-llines 36516  df-lplanes 36517  df-lvols 36518  df-lines 36519  df-psubsp 36521  df-pmap 36522  df-padd 36814  df-lhyp 37006  df-laut 37007  df-ldil 37122  df-ltrn 37123  df-trl 37177  df-tendo 37773  df-edring 37775  df-disoa 38047  df-dvech 38097  df-dib 38157  df-dic 38191  df-dih 38247
This theorem is referenced by:  dihopelvalc  38267
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