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Theorem dihopelvalcpre 38251
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 38239 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝑁‘(𝑋 𝑊))))
1312eleq2d 2903 . 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 1130 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 simp3l 1195 . . . 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 38196 . . . 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 1191 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
2019hllatd 36367 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21 simp2l 1193 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
22 simp1r 1192 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
231, 6lhpbase 37001 . . . . . 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 1365 . . . 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 1365 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) 𝑊)
291, 2, 6, 10, 8, 16diblss 38173 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (𝑁‘(𝑋 𝑊)) ∈ 𝑉)
3014, 26, 28, 29syl12anc 834 . . 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 38120 . . 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 1365 . 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 3503 . . . . . . . . 9 𝑥 ∈ V
39 vex 3503 . . . . . . . . 9 𝑦 ∈ V
402, 5, 6, 34, 35, 36, 9, 37, 38, 39dicopelval2 38184 . . . . . . . 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 38151 . . . . . . . 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 834 . . . . . . 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 1185 . . . . . . . . . . 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 37022 . . . . . . . . . . . . . . 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 1222 . . . . . . . . . . . . . 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 37582 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
5548, 52, 53, 54syl3anc 1365 . . . . . . . . . . . . 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 37770 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑦𝐸𝐺𝑇) → (𝑦𝐺) ∈ 𝑇)
5748, 50, 55, 56syl3anc 1365 . . . . . . . . . . . 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 2918 . . . . . . . . . . 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 37793 . . . . . . . . . . . . 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 2918 . . . . . . . . . . 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 2826 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
66 eqid 2826 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
676, 35, 36, 10, 65, 31, 66dvhopvadd 38096 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑦𝐸) ∧ (𝑧𝑇𝑤𝐸)) → (⟨𝑥, 𝑦+𝑧, 𝑤⟩) = ⟨(𝑥𝑧), (𝑦(+g‘(Scalar‘𝑈))𝑤)⟩)
6848, 58, 50, 60, 64, 67syl122anc 1373 . . . . . . . . . 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 38087 . . . . . . . . . . . . 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 7165 . . . . . . . . . . 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 4809 . . . . . . . . . 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 2861 . . . . . . . . 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 2837 . . . . . . . 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 5365 . . . . . . . . 9 (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
7961oveq2d 7164 . . . . . . . . . . . 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 37795 . . . . . . . . . . . . 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 2861 . . . . . . . . . . 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 2837 . . . . . . . . . 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 6669 . . . . . . . . . 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 2864 . . . . . . . . 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 2832 . . . . . . . . 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 6116 . . . . . . . . . . 11 (((𝑆𝐺) ∘ (𝑆𝐺)) ∘ 𝑧) = ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧))
95 simpl1 1185 . . . . . . . . . . . . . . 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 2918 . . . . . . . . . . . . . . . 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 37770 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝐺𝑇) → (𝑆𝐺) ∈ 𝑇)
10195, 98, 99, 100syl3anc 1365 . . . . . . . . . . . . . . 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 37127 . . . . . . . . . . . . . . 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 6640 . . . . . . . . . . . . . 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 5731 . . . . . . . . . . . 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 37127 . . . . . . . . . . . . . 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 6612 . . . . . . . . . . . . 13 (𝑧:𝐵1-1-onto𝐵𝑧:𝐵𝐵)
111 fcoi2 6550 . . . . . . . . . . . . 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 2862 . . . . . . . . . . 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 5731 . . . . . . . . . . . . . 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 2861 . . . . . . . . . . . . 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 5731 . . . . . . . . . . . 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 37149 . . . . . . . . . . . . . 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 37722 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇𝑧𝑇) → ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇)
12195, 101, 107, 120syl3anc 1365 . . . . . . . . . . . . 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 37741 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇) → ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧)) = (((𝑆𝐺) ∘ 𝑧) ∘ (𝑆𝐺)))
12395, 119, 121, 122syl3anc 1365 . . . . . . . . . . . 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 2864 . . . . . . . . . . 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 2887 . . . . . . . . . 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 1206 . . . . . . . . . . 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 1222 . . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 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 1365 . . . . . . . . . . . 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 2918 . . . . . . . . . . 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 37722 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥𝑇𝑧𝑇) → (𝑥𝑧) ∈ 𝑇)
145134, 142, 143, 144syl3anc 1365 . . . . . . . . . 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 2918 . . . . . . . . 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 1218 . . . . . . . . . . . 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 36367 . . . . . . . . . 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 37167 . . . . . . . . . . 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 1220 . . . . . . . . . . . 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 1219 . . . . . . . . . . . . 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 1365 . . . . . . . . . 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 1365 . . . . . . . . . 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 6669 . . . . . . . . . . 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 2864 . . . . . . . . . 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 1206 . . . . . . . . . . . 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 2919 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . . 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 1365 . . . . . . . . . . . . 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 37722 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇(𝑆𝐺) ∈ 𝑇) → (𝐹(𝑆𝐺)) ∈ 𝑇)
183174, 175, 181, 182syl3anc 1365 . . . . . . . . . . 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 2918 . . . . . . . . . 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 37187 . . . . . . . . . . . 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 36367 . . . . . . . . . . . 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 1366 . . . . . . . . . . 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 5732 . . . . . . . . . . . . . 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 37127 . . . . . . . . . . . . . . . 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 6612 . . . . . . . . . . . . . . 15 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
206 fcoi1 6549 . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . 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 6116 . . . . . . . . . . . . 13 ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)) = (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺)))
210208, 209syl6eqr 2879 . . . . . . . . . . . 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 37741 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ (𝐹(𝑆𝐺)) ∈ 𝑇) → ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))) = ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)))
212174, 180, 183, 211syl3anc 1365 . . . . . . . . . . . 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 2864 . . . . . . . . . . 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 5734 . . . . . . . . . . 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 2864 . . . . . . . . . 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 2832 . . . . . . . . . 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 1083 . . . . . 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 1920 . . 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 6680 . . . 4 (𝑆𝐺) ∈ V
226225cnvex 7618 . . . . 5 (𝑆𝐺) ∈ V
22776, 226coex 7623 . . . 4 (𝐹(𝑆𝐺)) ∈ V
22835fvexi 6681 . . . . . 6 𝑇 ∈ V
229228mptex 6981 . . . . 5 (𝑇 ↦ ( I ↾ 𝐵)) ∈ V
23043, 229eqeltri 2914 . . . 4 𝑍 ∈ V
231 biidd 263 . . . 4 (𝑥 = (𝑆𝐺) → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
232 eleq1 2905 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝐸𝑆𝐸))
233232anbi2d 628 . . . . 5 (𝑦 = 𝑆 → ((𝐹𝑇𝑦𝐸) ↔ (𝐹𝑇𝑆𝐸)))
234233anbi1d 629 . . . 4 (𝑦 = 𝑆 → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋)))
235 fveq2 6667 . . . . . 6 (𝑧 = (𝐹(𝑆𝐺)) → (𝑅𝑧) = (𝑅‘(𝐹(𝑆𝐺))))
236235breq1d 5073 . . . . 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 3552 . . 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 1081   = wceq 1530  wex 1773  wcel 2107  Vcvv 3500  cop 4570   class class class wbr 5063  cmpt 5143   I cid 5458  ccnv 5553  cres 5556  ccom 5558  wf 6348  1-1-ontowf1o 6351  cfv 6352  crio 7105  (class class class)co 7148  cmpo 7150  Basecbs 16473  +gcplusg 16555  Scalarcsca 16558  lecple 16562  occoc 16563  joincjn 17544  meetcmee 17545  Latclat 17645  LSSumclsm 18679  LSubSpclss 19623  Atomscatm 36266  HLchlt 36353  LHypclh 36987  LTrncltrn 37104  trLctrl 37161  TEndoctendo 37755  DVecHcdvh 38081  DIsoBcdib 38141  DIsoCcdic 38175  DIsoHcdih 38231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  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 35956
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  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 17890  df-mnd 17901  df-submnd 17945  df-grp 18036  df-minusg 18037  df-sbg 18038  df-subg 18206  df-cntz 18377  df-lsm 18681  df-cmn 18828  df-abl 18829  df-mgp 19160  df-ur 19172  df-ring 19219  df-oppr 19293  df-dvdsr 19311  df-unit 19312  df-invr 19342  df-dvr 19353  df-drng 19424  df-lmod 19556  df-lss 19624  df-lsp 19664  df-lvec 19795  df-oposet 36179  df-ol 36181  df-oml 36182  df-covers 36269  df-ats 36270  df-atl 36301  df-cvlat 36325  df-hlat 36354  df-llines 36501  df-lplanes 36502  df-lvols 36503  df-lines 36504  df-psubsp 36506  df-pmap 36507  df-padd 36799  df-lhyp 36991  df-laut 36992  df-ldil 37107  df-ltrn 37108  df-trl 37162  df-tendo 37758  df-edring 37760  df-disoa 38032  df-dvech 38082  df-dib 38142  df-dic 38176  df-dih 38232
This theorem is referenced by:  dihopelvalc  38252
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