Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dihopelvalcpre Structured version   Visualization version   GIF version

Theorem dihopelvalcpre 39711
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 39699 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝑁‘(𝑋 𝑊))))
1312eleq2d 2823 . 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 1136 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
15 simp3l 1201 . . . 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 39656 . . . 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 1197 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
2019hllatd 37826 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21 simp2l 1199 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑋𝐵)
22 simp1r 1198 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐻)
231, 6lhpbase 38461 . . . . . 6 (𝑊𝐻𝑊𝐵)
2422, 23syl 17 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → 𝑊𝐵)
251, 4latmcl 18329 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) ∈ 𝐵)
2620, 21, 24, 25syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) ∈ 𝐵)
271, 2, 4latmle2 18354 . . . . 5 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑊)
2820, 21, 24, 27syl3anc 1371 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝑋 𝑊) 𝑊)
291, 2, 6, 10, 8, 16diblss 39633 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋 𝑊) ∈ 𝐵 ∧ (𝑋 𝑊) 𝑊)) → (𝑁‘(𝑋 𝑊)) ∈ 𝑉)
3014, 26, 28, 29syl12anc 835 . . 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 39580 . . 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 1371 . 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 3449 . . . . . . . . 9 𝑥 ∈ V
39 vex 3449 . . . . . . . . 9 𝑦 ∈ V
402, 5, 6, 34, 35, 36, 9, 37, 38, 39dicopelval2 39644 . . . . . . . 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 39611 . . . . . . . 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 835 . . . . . . 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 631 . . . . . 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 630 . . . . 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 1191 . . . . . . . . . . 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 777 . . . . . . . . . . . 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 778 . . . . . . . . . . . . 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 38482 . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . 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 39042 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝐺𝑇)
5548, 52, 53, 54syl3anc 1371 . . . . . . . . . . . . 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 39230 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑦𝐸𝐺𝑇) → (𝑦𝐺) ∈ 𝑇)
5748, 50, 55, 56syl3anc 1371 . . . . . . . . . . . 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 2838 . . . . . . . . . . 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 777 . . . . . . . . . . . 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 780 . . . . . . . . . . . 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 39253 . . . . . . . . . . . . 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 2838 . . . . . . . . . . 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 2736 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
66 eqid 2736 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
676, 35, 36, 10, 65, 31, 66dvhopvadd 39556 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑦𝐸) ∧ (𝑧𝑇𝑤𝐸)) → (⟨𝑥, 𝑦+𝑧, 𝑤⟩) = ⟨(𝑥𝑧), (𝑦(+g‘(Scalar‘𝑈))𝑤)⟩)
6848, 58, 50, 60, 64, 67syl122anc 1379 . . . . . . . . . 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 39547 . . . . . . . . . . . . 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 7374 . . . . . . . . . . 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 4837 . . . . . . . . . 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 2776 . . . . . . . . 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 2747 . . . . . . . 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 5433 . . . . . . . . 9 (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
7961oveq2d 7373 . . . . . . . . . . . 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 39255 . . . . . . . . . . . . 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 2776 . . . . . . . . . . 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 2747 . . . . . . . . . 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 629 . . . . . . . . 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 282 . . . . . . . 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 278 . . . . . . 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 773 . . . . . . . . . . 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 780 . . . . . . . . . . 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 6844 . . . . . . . . . 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 2779 . . . . . . . . 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 2742 . . . . . . . . 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 6217 . . . . . . . . . . 11 (((𝑆𝐺) ∘ (𝑆𝐺)) ∘ 𝑧) = ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧))
95 simpl1 1191 . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . 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 39230 . . . . . . . . . . . . . . . 16 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝐺𝑇) → (𝑆𝐺) ∈ 𝑇)
10195, 98, 99, 100syl3anc 1371 . . . . . . . . . . . . . . 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 38587 . . . . . . . . . . . . . . 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 6813 . . . . . . . . . . . . . 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 5817 . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 38587 . . . . . . . . . . . . . 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 6784 . . . . . . . . . . . . 13 (𝑧:𝐵1-1-onto𝐵𝑧:𝐵𝐵)
111 fcoi2 6717 . . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 779 . . . . . . . . . . . . . 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 5817 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 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 5817 . . . . . . . . . . . 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 38609 . . . . . . . . . . . . . 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 39182 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇𝑧𝑇) → ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇)
12195, 101, 107, 120syl3anc 1371 . . . . . . . . . . . . 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 39201 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ ((𝑆𝐺) ∘ 𝑧) ∈ 𝑇) → ((𝑆𝐺) ∘ ((𝑆𝐺) ∘ 𝑧)) = (((𝑆𝐺) ∘ 𝑧) ∘ (𝑆𝐺)))
12395, 119, 121, 122syl3anc 1371 . . . . . . . . . . . 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 2779 . . . . . . . . . . 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 2802 . . . . . . . . . 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 776 . . . . . . . . . . 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 278 . . . . 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 779 . . . . . . . . . 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 1212 . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 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 1371 . . . . . . . . . . . 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 2838 . . . . . . . . . . 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 726 . . . . . . . . . . 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 39182 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑥𝑇𝑧𝑇) → (𝑥𝑧) ∈ 𝑇)
145134, 142, 143, 144syl3anc 1371 . . . . . . . . . 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 2838 . . . . . . . . 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 1224 . . . . . . . . . . . 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 37826 . . . . . . . . . 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 38627 . . . . . . . . . . 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 1226 . . . . . . . . . . . 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 1225 . . . . . . . . . . . . 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 1371 . . . . . . . . . 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 778 . . . . . . . . . . 11 (((𝑥 = (𝑦𝐺) ∧ 𝑦𝐸) ∧ ((𝑧𝑇 ∧ (𝑅𝑧) (𝑋 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅𝑧) (𝑋 𝑊))
159158ad2antrl 726 . . . . . . . . . 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 18353 . . . . . . . . . . 11 ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑊𝐵) → (𝑋 𝑊) 𝑋)
161149, 153, 156, 160syl3anc 1371 . . . . . . . . . 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 18335 . . . . . . . . 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 777 . . . . . . . . . . . 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 778 . . . . . . . . . . . . 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 6844 . . . . . . . . . . 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 2779 . . . . . . . . . 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 778 . . . . . . . . . 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 779 . . . . . . . . . . . 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 1212 . . . . . . . . . . . 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 777 . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . 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 39182 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇(𝑆𝐺) ∈ 𝑇) → (𝐹(𝑆𝐺)) ∈ 𝑇)
183174, 175, 181, 182syl3anc 1371 . . . . . . . . . . 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 2838 . . . . . . . . . 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 771 . . . . . . . . . . 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 38647 . . . . . . . . . . . 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 37826 . . . . . . . . . . . 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 18355 . . . . . . . . . . . 12 ((𝐾 ∈ Lat ∧ ((𝑅𝑧) ∈ 𝐵𝑋𝐵𝑊𝐵)) → (((𝑅𝑧) 𝑋 ∧ (𝑅𝑧) 𝑊) ↔ (𝑅𝑧) (𝑋 𝑊)))
195189, 190, 191, 193, 194syl13anc 1372 . . . . . . . . . . 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 710 . . . . . . . . . 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 780 . . . . . . . . . . 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 5818 . . . . . . . . . . . . . 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 38587 . . . . . . . . . . . . . . . 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 6784 . . . . . . . . . . . . . . 15 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
206 fcoi1 6716 . . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . . 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 6217 . . . . . . . . . . . . 13 ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)) = (𝐹 ∘ ((𝑆𝐺) ∘ (𝑆𝐺)))
210208, 209eqtr4di 2794 . . . . . . . . . . . 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 39201 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐺) ∈ 𝑇 ∧ (𝐹(𝑆𝐺)) ∈ 𝑇) → ((𝑆𝐺) ∘ (𝐹(𝑆𝐺))) = ((𝐹(𝑆𝐺)) ∘ (𝑆𝐺)))
212174, 180, 183, 211syl3anc 1371 . . . . . . . . . . . 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 2779 . . . . . . . . . . 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 5820 . . . . . . . . . . 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 2779 . . . . . . . . . 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 2742 . . . . . . . . . 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 799 . . . . . . 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 288 . . . . 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 304 . . . 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 1929 . . 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 6855 . . . 4 (𝑆𝐺) ∈ V
226225cnvex 7862 . . . . 5 (𝑆𝐺) ∈ V
22776, 226coex 7867 . . . 4 (𝐹(𝑆𝐺)) ∈ V
22835fvexi 6856 . . . . . 6 𝑇 ∈ V
229228mptex 7173 . . . . 5 (𝑇 ↦ ( I ↾ 𝐵)) ∈ V
23043, 229eqeltri 2834 . . . 4 𝑍 ∈ V
231 biidd 261 . . . 4 (𝑥 = (𝑆𝐺) → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)))
232 eleq1 2825 . . . . . 6 (𝑦 = 𝑆 → (𝑦𝐸𝑆𝐸))
233232anbi2d 629 . . . . 5 (𝑦 = 𝑆 → ((𝐹𝑇𝑦𝐸) ↔ (𝐹𝑇𝑆𝐸)))
234233anbi1d 630 . . . 4 (𝑦 = 𝑆 → (((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋)))
235 fveq2 6842 . . . . . 6 (𝑧 = (𝐹(𝑆𝐺)) → (𝑅𝑧) = (𝑅‘(𝐹(𝑆𝐺))))
236235breq1d 5115 . . . . 5 (𝑧 = (𝐹(𝑆𝐺)) → ((𝑅𝑧) 𝑋 ↔ (𝑅‘(𝐹(𝑆𝐺))) 𝑋))
237236anbi2d 629 . . . 4 (𝑧 = (𝐹(𝑆𝐺)) → (((𝐹𝑇𝑆𝐸) ∧ (𝑅𝑧) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
238 biidd 261 . . . 4 (𝑤 = 𝑍 → (((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋)))
239225, 77, 227, 230, 231, 234, 237, 238ceqsex4v 3501 . . 3 (∃𝑥𝑦𝑧𝑤((𝑥 = (𝑆𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹(𝑆𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹𝑇𝑦𝐸) ∧ (𝑅𝑧) 𝑋)) ↔ ((𝐹𝑇𝑆𝐸) ∧ (𝑅‘(𝐹(𝑆𝐺))) 𝑋))
240224, 239bitrdi 286 . 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 304 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  Vcvv 3445  cop 4592   class class class wbr 5105  cmpt 5188   I cid 5530  ccnv 5632  cres 5635  ccom 5637  wf 6492  1-1-ontowf1o 6495  cfv 6496  crio 7312  (class class class)co 7357  cmpo 7359  Basecbs 17083  +gcplusg 17133  Scalarcsca 17136  lecple 17140  occoc 17141  joincjn 18200  meetcmee 18201  Latclat 18320  LSSumclsm 19416  LSubSpclss 20392  Atomscatm 37725  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  trLctrl 38621  TEndoctendo 39215  DVecHcdvh 39541  DIsoBcdib 39601  DIsoCcdic 39635  DIsoHcdih 39691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-riotaBAD 37415
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-undef 8204  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-0g 17323  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-p1 18315  df-lat 18321  df-clat 18388  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-subg 18925  df-cntz 19097  df-lsm 19418  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-dvr 20112  df-drng 20187  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lvec 20564  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962  df-lvols 37963  df-lines 37964  df-psubsp 37966  df-pmap 37967  df-padd 38259  df-lhyp 38451  df-laut 38452  df-ldil 38567  df-ltrn 38568  df-trl 38622  df-tendo 39218  df-edring 39220  df-disoa 39492  df-dvech 39542  df-dib 39602  df-dic 39636  df-dih 39692
This theorem is referenced by:  dihopelvalc  39712
  Copyright terms: Public domain W3C validator