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Theorem dihopelvalcpre 41250
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 41238 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝑁‘(𝑋 𝑊))))
1312eleq2d 2827 . 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 1137 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 simp3l 1202 . . . 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 41195 . . . 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 1198 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
2019hllatd 39365 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21 simp2l 1200 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
22 simp1r 1199 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
231, 6lhpbase 40000 . . . . . 6 (𝑊𝐻𝑊𝐵)
2422, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
251, 4latmcl 18485 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
2620, 21, 24, 25syl3anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) ∈ 𝐵)
271, 2, 4latmle2 18510 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
2820, 21, 24, 27syl3anc 1373 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) 𝑊)
291, 2, 6, 10, 8, 16diblss 41172 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (𝑁‘(𝑋 𝑊)) ∈ 𝑉)
3014, 26, 28, 29syl12anc 837 . . 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 41119 . . 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 1373 . 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 3484 . . . . . . . . 9 𝑥 ∈ V
39 vex 3484 . . . . . . . . 9 𝑦 ∈ V
402, 5, 6, 34, 35, 36, 9, 37, 38, 39dicopelval2 41183 . . . . . . . 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 41150 . . . . . . . 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 837 . . . . . . 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 632 . . . . . 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 631 . . . . 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 1192 . . . . . . . . . . 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 779 . . . . . . . . . . . 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 780 . . . . . . . . . . . . 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 40021 . . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . 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 40581 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
5548, 52, 53, 54syl3anc 1373 . . . . . . . . . . . . 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 40769 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑦𝐸𝐺𝑇) → (𝑦𝐺) ∈ 𝑇)
5748, 50, 55, 56syl3anc 1373 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 779 . . . . . . . . . . . 12 (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) → 𝑧𝑇)
6059adantl 481 . . . . . . . . . . 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 782 . . . . . . . . . . . 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 40792 . . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
66 eqid 2737 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
676, 35, 36, 10, 65, 31, 66dvhopvadd 41095 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑦𝐸) ∧ (𝑧𝑇𝑤𝐸)) → (⟨𝑥, 𝑦+𝑧, 𝑤⟩) = ⟨(𝑥𝑧), (𝑦(+g‘(Scalar‘𝑈))𝑤)⟩)
6848, 58, 50, 60, 64, 67syl122anc 1381 . . . . . . . . . 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 41086 . . . . . . . . . . . . 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 7448 . . . . . . . . . . 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 4880 . . . . . . . . . 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 2777 . . . . . . . . 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 2748 . . . . . . . 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 5481 . . . . . . . . 9 (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
7961oveq2d 7447 . . . . . . . . . . . 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 40794 . . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 2748 . . . . . . . . . 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 630 . . . . . . . . 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, 84bitrid 283 . . . . . . . 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 279 . . . . . . 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 775 . . . . . . . . . . 11 ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → 𝑥 = (𝑦𝐺))
8988adantl 481 . . . . . . . . . 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 782 . . . . . . . . . . 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 6908 . . . . . . . . . 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 2780 . . . . . . . . 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 2743 . . . . . . . . 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 6285 . . . . . . . . . . 11 (((𝑆𝐺) ∘ (𝑆𝐺)) ∘ 𝑧) = ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧))
95 simpl1 1192 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . 18 ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → 𝑦𝐸)
9796adantl 481 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . 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 40769 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝐺𝑇) → (𝑆𝐺) ∈ 𝑇)
10195, 98, 99, 100syl3anc 1373 . . . . . . . . . . . . . . 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 40126 . . . . . . . . . . . . . . 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 6877 . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . 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 728 . . . . . . . . . . . . . 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 40126 . . . . . . . . . . . . . 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 6848 . . . . . . . . . . . . 13 (𝑧:𝐵1-1-onto𝐵𝑧:𝐵𝐵)
111 fcoi2 6783 . . . . . . . . . . . . 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 2778 . . . . . . . . . . 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 781 . . . . . . . . . . . . . 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 5872 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 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 5872 . . . . . . . . . . . 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 40148 . . . . . . . . . . . . . 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 40721 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇𝑧𝑇) → ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇)
12195, 101, 107, 120syl3anc 1373 . . . . . . . . . . . . 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 40740 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇) → ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧)) = (((𝑆𝐺) ∘ 𝑧) ∘ (𝑆𝐺)))
12395, 119, 121, 122syl3anc 1373 . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 2803 . . . . . . . . . 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 778 . . . . . . . . . . 11 ((((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥𝑧) ∧ 𝑆 = 𝑦)) → 𝑤 = 𝑍)
127126adantl 481 . . . . . . . . . 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 511 . . . . . . . . 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 514 . . . . . . . 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 412 . . . . . . 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 562 . . . . . 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 279 . . . . 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 781 . . . . . . . . . 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 1213 . . . . . . . . . . 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 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))) → 𝑥 = (𝑦𝐺))
13696adantl 481 . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 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 1373 . . . . . . . . . . . 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 2841 . . . . . . . . . . 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 728 . . . . . . . . . . 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 40721 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥𝑇𝑧𝑇) → (𝑥𝑧) ∈ 𝑇)
145134, 142, 143, 144syl3anc 1373 . . . . . . . . . 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 2841 . . . . . . . . 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 1225 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 39365 . . . . . . . . . 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 40166 . . . . . . . . . . 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 1227 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 1226 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 1373 . . . . . . . . . 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 780 . . . . . . . . . . 11 (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅𝑧) (𝑋 𝑊))
159158ad2antrl 728 . . . . . . . . . 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 18509 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑋)
161149, 153, 156, 160syl3anc 1373 . . . . . . . . . 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 18491 . . . . . . . . 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 514 . . . . . . . 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 779 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 780 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 6908 . . . . . . . . . . 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 2780 . . . . . . . . . 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 780 . . . . . . . . . 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 511 . . . . . . . . 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 781 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 1213 . . . . . . . . . . . 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 779 . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . 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 40721 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇(𝑆𝐺) ∈ 𝑇) → (𝐹(𝑆𝐺)) ∈ 𝑇)
183174, 175, 181, 182syl3anc 1373 . . . . . . . . . . 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 2841 . . . . . . . . . 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 773 . . . . . . . . . . 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 40186 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 39365 . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 18511 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ ((𝑅𝑧) ∈ 𝐵𝑋𝐵𝑊𝐵)) → (((𝑅𝑧) 𝑋 ∧ (𝑅𝑧) 𝑊) ↔ (𝑅𝑧) (𝑋 𝑊)))
195189, 190, 191, 193, 194syl13anc 1374 . . . . . . . . . . 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 712 . . . . . . . . . 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 782 . . . . . . . . . . 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 480 . . . . . . . . . 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 514 . . . . . . . . 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 5873 . . . . . . . . . . . . . 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 40126 . . . . . . . . . . . . . . . 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 6848 . . . . . . . . . . . . . . 15 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
206 fcoi1 6782 . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . 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 6285 . . . . . . . . . . . . 13 ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)) = (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺)))
210208, 209eqtr4di 2795 . . . . . . . . . . . 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 40740 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ (𝐹(𝑆𝐺)) ∈ 𝑇) → ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))) = ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)))
212174, 180, 183, 211syl3anc 1373 . . . . . . . . . . . 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 2780 . . . . . . . . . . 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 5875 . . . . . . . . . . 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 2780 . . . . . . . . . 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 2743 . . . . . . . . . 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 511 . . . . . . . . 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 514 . . . . . . . 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 801 . . . . . . 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 1089 . . . . . 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, 221bitr4di 289 . . . . 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 305 . . . 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 1926 . . 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 6919 . . . 4 (𝑆𝐺) ∈ V
226225cnvex 7947 . . . . 5 (𝑆𝐺) ∈ V
22776, 226coex 7952 . . . 4 (𝐹(𝑆𝐺)) ∈ V
22835fvexi 6920 . . . . . 6 𝑇 ∈ V
229228mptex 7243 . . . . 5 (𝑇 ↦ ( I ↾ 𝐵)) ∈ V
23043, 229eqeltri 2837 . . . 4 𝑍 ∈ V
231 biidd 262 . . . 4 (𝑥 = (𝑆𝐺) → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
232 eleq1 2829 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝐸𝑆𝐸))
233232anbi2d 630 . . . . 5 (𝑦 = 𝑆 → ((𝐹𝑇𝑦𝐸) ↔ (𝐹𝑇𝑆𝐸)))
234233anbi1d 631 . . . 4 (𝑦 = 𝑆 → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋)))
235 fveq2 6906 . . . . . 6 (𝑧 = (𝐹(𝑆𝐺)) → (𝑅𝑧) = (𝑅‘(𝐹(𝑆𝐺))))
236235breq1d 5153 . . . . 5 (𝑧 = (𝐹(𝑆𝐺)) → ((𝑅𝑧) 𝑋 ↔ (𝑅‘(𝐹(𝑆𝐺))) 𝑋))
237236anbi2d 630 . . . 4 (𝑧 = (𝐹(𝑆𝐺)) → (((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
238 biidd 262 . . . 4 (𝑤 = 𝑍 → (((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
239225, 77, 227, 230, 231, 234, 237, 238ceqsex4v 3538 . . 3 (∃𝑥𝑦𝑧𝑤((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋))
240224, 239bitrdi 287 . 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 305 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577  ccnv 5684  cres 5687  ccom 5689  wf 6557  1-1-ontowf1o 6560  cfv 6561  crio 7387  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  Latclat 18476  LSSumclsm 19652  LSubSpclss 20929  Atomscatm 39264  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  TEndoctendo 40754  DVecHcdvh 41080  DIsoBcdib 41140  DIsoCcdic 41174  DIsoHcdih 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-undef 8298  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-cntz 19335  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-drng 20731  df-lmod 20860  df-lss 20930  df-lsp 20970  df-lvec 21102  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tendo 40757  df-edring 40759  df-disoa 41031  df-dvech 41081  df-dib 41141  df-dic 41175  df-dih 41231
This theorem is referenced by:  dihopelvalc  41251
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