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Theorem dihopelvalcpre 38264
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 38252 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝑁‘(𝑋 𝑊))))
1312eleq2d 2895 . 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 38209 . . . 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 36380 . . . . 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 37014 . . . . . 6 (𝑊𝐻𝑊𝐵)
2422, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
251, 4latmcl 17650 . . . . 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 17675 . . . . 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 38186 . . . 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 38133 . . 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 3495 . . . . . . . . 9 𝑥 ∈ V
39 vex 3495 . . . . . . . . 9 𝑦 ∈ V
402, 5, 6, 34, 35, 36, 9, 37, 38, 39dicopelval2 38197 . . . . . . . 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 38164 . . . . . . . 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 37035 . . . . . . . . . . . . . . 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 37595 . . . . . . . . . . . . . 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 37783 . . . . . . . . . . . . 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 2910 . . . . . . . . . . 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 37806 . . . . . . . . . . . . 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 2910 . . . . . . . . . . 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 2818 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
66 eqid 2818 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
676, 35, 36, 10, 65, 31, 66dvhopvadd 38109 . . . . . . . . . . 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 38100 . . . . . . . . . . . . 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 4802 . . . . . . . . . 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 2853 . . . . . . . . 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 2829 . . . . . . . 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 5359 . . . . . . . . 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 37808 . . . . . . . . . . . . 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 2853 . . . . . . . . . . 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 2829 . . . . . . . . . 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 6665 . . . . . . . . . 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 2856 . . . . . . . . 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 2824 . . . . . . . . 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 6111 . . . . . . . . . . 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 2910 . . . . . . . . . . . . . . . 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 37783 . . . . . . . . . . . . . . . 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 37140 . . . . . . . . . . . . . . 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 6636 . . . . . . . . . . . . . 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 5725 . . . . . . . . . . . 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 37140 . . . . . . . . . . . . . 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 6608 . . . . . . . . . . . . 13 (𝑧:𝐵1-1-onto𝐵𝑧:𝐵𝐵)
111 fcoi2 6546 . . . . . . . . . . . . 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 2854 . . . . . . . . . . 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 5725 . . . . . . . . . . . . . 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 2853 . . . . . . . . . . . . 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 5725 . . . . . . . . . . . 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 37162 . . . . . . . . . . . . . 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 37735 . . . . . . . . . . . . . 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 37754 . . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 2879 . . . . . . . . . 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 2910 . . . . . . . . . . 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 37735 . . . . . . . . . . 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 2910 . . . . . . . . 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 36380 . . . . . . . . . 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 37180 . . . . . . . . . . 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 17674 . . . . . . . . . . 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 17656 . . . . . . . . 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 6665 . . . . . . . . . . 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 2856 . . . . . . . . . 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 2911 . . . . . . . . . . . . . 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 37735 . . . . . . . . . . . 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 2910 . . . . . . . . . 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 37200 . . . . . . . . . . . 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 36380 . . . . . . . . . . . 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 17676 . . . . . . . . . . . 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 5726 . . . . . . . . . . . . . 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 37140 . . . . . . . . . . . . . . . 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 6608 . . . . . . . . . . . . . . 15 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
206 fcoi1 6545 . . . . . . . . . . . . . . 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 2854 . . . . . . . . . . . . 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 6111 . . . . . . . . . . . . 13 ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)) = (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺)))
210208, 209syl6eqr 2871 . . . . . . . . . . . 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 37754 . . . . . . . . . . . . 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 2856 . . . . . . . . . . 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 5728 . . . . . . . . . . 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 2856 . . . . . . . . . 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 2824 . . . . . . . . . 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 6676 . . . 4 (𝑆𝐺) ∈ V
226225cnvex 7619 . . . . 5 (𝑆𝐺) ∈ V
22776, 226coex 7624 . . . 4 (𝐹(𝑆𝐺)) ∈ V
22835fvexi 6677 . . . . . 6 𝑇 ∈ V
229228mptex 6977 . . . . 5 (𝑇 ↦ ( I ↾ 𝐵)) ∈ V
23043, 229eqeltri 2906 . . . 4 𝑍 ∈ V
231 biidd 263 . . . 4 (𝑥 = (𝑆𝐺) → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
232 eleq1 2897 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝐸𝑆𝐸))
233232anbi2d 628 . . . . 5 (𝑦 = 𝑆 → ((𝐹𝑇𝑦𝐸) ↔ (𝐹𝑇𝑆𝐸)))
234233anbi1d 629 . . . 4 (𝑦 = 𝑆 → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋)))
235 fveq2 6663 . . . . . 6 (𝑧 = (𝐹(𝑆𝐺)) → (𝑅𝑧) = (𝑅‘(𝐹(𝑆𝐺))))
236235breq1d 5067 . . . . 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 3544 . . 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 3492  cop 4563   class class class wbr 5057  cmpt 5137   I cid 5452  ccnv 5547  cres 5550  ccom 5552  wf 6344  1-1-ontowf1o 6347  cfv 6348  crio 7102  (class class class)co 7145  cmpo 7147  Basecbs 16471  +gcplusg 16553  Scalarcsca 16556  lecple 16560  occoc 16561  joincjn 17542  meetcmee 17543  Latclat 17643  LSSumclsm 18688  LSubSpclss 19632  Atomscatm 36279  HLchlt 36366  LHypclh 37000  LTrncltrn 37117  trLctrl 37174  TEndoctendo 37768  DVecHcdvh 38094  DIsoBcdib 38154  DIsoCcdic 38188  DIsoHcdih 38244
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 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-riotaBAD 35969
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 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-tpos 7881  df-undef 7928  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-0g 16703  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-p1 17638  df-lat 17644  df-clat 17706  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-grp 18044  df-minusg 18045  df-sbg 18046  df-subg 18214  df-cntz 18385  df-lsm 18690  df-cmn 18837  df-abl 18838  df-mgp 19169  df-ur 19181  df-ring 19228  df-oppr 19302  df-dvdsr 19320  df-unit 19321  df-invr 19351  df-dvr 19362  df-drng 19433  df-lmod 19565  df-lss 19633  df-lsp 19673  df-lvec 19804  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-llines 36514  df-lplanes 36515  df-lvols 36516  df-lines 36517  df-psubsp 36519  df-pmap 36520  df-padd 36812  df-lhyp 37004  df-laut 37005  df-ldil 37120  df-ltrn 37121  df-trl 37175  df-tendo 37771  df-edring 37773  df-disoa 38045  df-dvech 38095  df-dib 38155  df-dic 38189  df-dih 38245
This theorem is referenced by:  dihopelvalc  38265
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