dihopelvalcpre - Mathbox for Norm Megill

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Theorem dihopelvalcpre 35555

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 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
dihopelvalcp.f	⊢ 𝐹 ∈ V
dihopelvalcp.s	⊢ 𝑆 ∈ V
dihopelvalcp.z	⊢ 𝑍 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihopelvalcp.n	⊢ 𝑁 = ((DIsoB‘𝐾)‘𝑊)
dihopelvalcp.c	⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihopelvalcp.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihopelvalcp.d	⊢ + = (+_g‘𝑈)
dihopelvalcp.v	⊢ 𝑉 = (LSubSp‘𝑈)
dihopelvalcp.y	⊢ ⊕ = (LSSum‘𝑈)
dihopelvalcp.o	⊢ 𝑂 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑎‘ℎ) ∘ (𝑏‘ℎ))))

**Assertion**
Ref	Expression
dihopelvalcpre	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))

Distinct variable groups: ≤ ,𝑔 𝐴,𝑔 𝑃,𝑔 𝑎,𝑏,𝐸 𝑔,ℎ,𝐻 𝑔,𝑎,ℎ,𝐾,𝑏 𝐵,ℎ 𝑇,𝑎,𝑏,𝑔,ℎ 𝑊,𝑎,𝑏,𝑔,ℎ 𝑄,𝑔 Allowed substitution hints: 𝐴(ℎ,𝑎,𝑏) 𝐵(𝑔,𝑎,𝑏) 𝐶(𝑔,ℎ,𝑎,𝑏) 𝑃(ℎ,𝑎,𝑏) + (𝑔,ℎ,𝑎,𝑏) ⊕ (𝑔,ℎ,𝑎,𝑏) 𝑄(ℎ,𝑎,𝑏) 𝑅(𝑔,ℎ,𝑎,𝑏) 𝑆(𝑔,ℎ,𝑎,𝑏) 𝑈(𝑔,ℎ,𝑎,𝑏) 𝐸(𝑔,ℎ) 𝐹(𝑔,ℎ,𝑎,𝑏) 𝐺(𝑔,ℎ,𝑎,𝑏) 𝐻(𝑎,𝑏) 𝐼(𝑔,ℎ,𝑎,𝑏) ∨ (𝑔,ℎ,𝑎,𝑏) ≤ (ℎ,𝑎,𝑏) ∧ (𝑔,ℎ,𝑎,𝑏) 𝑁(𝑔,ℎ,𝑎,𝑏) 𝑂(𝑔,ℎ,𝑎,𝑏) 𝑉(𝑔,ℎ,𝑎,𝑏) 𝑋(𝑔,ℎ,𝑎,𝑏) 𝑍(𝑔,ℎ,𝑎,𝑏)

Proof of Theorem dihopelvalcpre Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	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)
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	dihvalcq 35543	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐼‘𝑋) = ((𝐶‘𝑄) ⊕ (𝑁‘(𝑋 ∧ 𝑊))))
13	12	eleq2d 2673	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ⟨𝐹, 𝑆⟩ ∈ ((𝐶‘𝑄) ⊕ (𝑁‘(𝑋 ∧ 𝑊)))))
14		simp1 1054	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3l 1082	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
16		dihopelvalcp.v	. . . . 5 ⊢ 𝑉 = (LSubSp‘𝑈)
17	2, 5, 6, 10, 9, 16	diclss 35500	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐶‘𝑄) ∈ 𝑉)
18	14, 15, 17	syl2anc 691	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐶‘𝑄) ∈ 𝑉)
19		simp1l 1078	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
20		hllat 33668	. . . . . 6 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
21	19, 20	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
22		simp2l 1080	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
23		simp1r 1079	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
24	1, 6	lhpbase 34302	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
25	23, 24	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵)
26	1, 4	latmcl 16875	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
27	21, 22, 25, 26	syl3anc 1318	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
28	1, 2, 4	latmle2 16900	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
29	21, 22, 25, 28	syl3anc 1318	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∧ 𝑊) ≤ 𝑊)
30	1, 2, 6, 10, 8, 16	diblss 35477	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉)
31	14, 27, 29, 30	syl12anc 1316	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉)
32		dihopelvalcp.d	. . . 4 ⊢ + = (+_g‘𝑈)
33	6, 10, 32, 16, 11	dvhopellsm 35424	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐶‘𝑄) ∈ 𝑉 ∧ (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉) → (⟨𝐹, 𝑆⟩ ∈ ((𝐶‘𝑄) ⊕ (𝑁‘(𝑋 ∧ 𝑊))) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))))
34	14, 18, 31, 33	syl3anc 1318	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ ((𝐶‘𝑄) ⊕ (𝑁‘(𝑋 ∧ 𝑊))) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))))
35		dihopelvalcp.p	. . . . . . . . 9 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
36		dihopelvalcp.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
37		dihopelvalcp.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
38		dihopelvalcp.g	. . . . . . . . 9 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
39		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
40		vex 3176	. . . . . . . . 9 ⊢ 𝑦 ∈ V
41	2, 5, 6, 35, 36, 37, 9, 38, 39, 40	dicopelval2 35488	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ↔ (𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸)))
42	14, 15, 41	syl2anc 691	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ↔ (𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸)))
43		dihopelvalcp.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
44		dihopelvalcp.z	. . . . . . . . 9 ⊢ 𝑍 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
45	1, 2, 6, 36, 43, 44, 8	dibopelval3 35455	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊)) ↔ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)))
46	14, 27, 29, 45	syl12anc 1316	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊)) ↔ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)))
47	42, 46	anbi12d 743	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ↔ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))))
48	47	anbi1d 737	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩)) ↔ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))))
49		simpl1 1057	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
50		simprll 798	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑥 = (𝑦‘𝐺))
51		simprlr 799	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑦 ∈ 𝐸)
52	2, 5, 6, 35	lhpocnel2 34323	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
53	49, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
54		simpl3l 1109	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
55	2, 5, 6, 36, 38	ltrniotacl 34885	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
56	49, 53, 54, 55	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝐺 ∈ 𝑇)
57	6, 36, 37	tendocl 35073	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑦‘𝐺) ∈ 𝑇)
58	49, 51, 56, 57	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦‘𝐺) ∈ 𝑇)
59	50, 58	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑥 ∈ 𝑇)
60		simprll 798	. . . . . . . . . . . 12 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → 𝑧 ∈ 𝑇)
61	60	adantl 481	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑧 ∈ 𝑇)
62		simprrr 801	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑤 = 𝑍)
63	1, 6, 36, 37, 44	tendo0cl 35096	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑍 ∈ 𝐸)
64	49, 63	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑍 ∈ 𝐸)
65	62, 64	eqeltrd 2688	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → 𝑤 ∈ 𝐸)
66		eqid 2610	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
67		eqid 2610	. . . . . . . . . . . 12 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
68	6, 36, 37, 10, 66, 32, 67	dvhopvadd 35400	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑧 ∈ 𝑇 ∧ 𝑤 ∈ 𝐸)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 ∘ 𝑧), (𝑦(+_g‘(Scalar‘𝑈))𝑤)⟩)
69	49, 59, 51, 61, 65, 68	syl122anc 1327	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 ∘ 𝑧), (𝑦(+_g‘(Scalar‘𝑈))𝑤)⟩)
70		dihopelvalcp.o	. . . . . . . . . . . . . 14 ⊢ 𝑂 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑎‘ℎ) ∘ (𝑏‘ℎ))))
71	6, 36, 37, 10, 66, 70, 67	dvhfplusr 35391	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘(Scalar‘𝑈)) = 𝑂)
72	49, 71	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (+_g‘(Scalar‘𝑈)) = 𝑂)
73	72	oveqd 6566	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦(+_g‘(Scalar‘𝑈))𝑤) = (𝑦𝑂𝑤))
74	73	opeq2d 4347	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → ⟨(𝑥 ∘ 𝑧), (𝑦(+_g‘(Scalar‘𝑈))𝑤)⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩)
75	69, 74	eqtrd 2644	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩)
76	75	eqeq2d 2620	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩))
77		dihopelvalcp.f	. . . . . . . . . 10 ⊢ 𝐹 ∈ V
78		dihopelvalcp.s	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
79	77, 78	opth 4871	. . . . . . . . 9 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
80	62	oveq2d 6565	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝑂𝑤) = (𝑦𝑂𝑍))
81	1, 6, 36, 37, 44, 70	tendo0plr 35098	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸) → (𝑦𝑂𝑍) = 𝑦)
82	49, 51, 81	syl2anc 691	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝑂𝑍) = 𝑦)
83	80, 82	eqtrd 2644	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑦𝑂𝑤) = 𝑦)
84	83	eqeq2d 2620	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (𝑆 = (𝑦𝑂𝑤) ↔ 𝑆 = 𝑦))
85	84	anbi2d 736	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → ((𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)) ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)))
86	79, 85	syl5bb 271	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)))
87	76, 86	bitrd 267	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)))
88	87	pm5.32da 671	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩)) ↔ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))))
89		simplll 794	. . . . . . . . . . 11 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑥 = (𝑦‘𝐺))
90	89	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 = (𝑦‘𝐺))
91		simprrr 801	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑆 = 𝑦)
92	91	fveq1d 6105	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑆‘𝐺) = (𝑦‘𝐺))
93	90, 92	eqtr4d 2647	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 = (𝑆‘𝐺))
94	91	eqcomd 2616	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑦 = 𝑆)
95		coass 5571	. . . . . . . . . . 11 ⊢ ((^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) ∘ 𝑧) = (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧))
96		simpl1 1057	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
97		simpllr 795	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑦 ∈ 𝐸)
98	97	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑦 ∈ 𝐸)
99	91, 98	eqeltrd 2688	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑆 ∈ 𝐸)
100	56	adantrr 749	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐺 ∈ 𝑇)
101	6, 36, 37	tendocl 35073	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑆‘𝐺) ∈ 𝑇)
102	96, 99, 100, 101	syl3anc 1318	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑆‘𝐺) ∈ 𝑇)
103	1, 6, 36	ltrn1o 34428	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → (𝑆‘𝐺):𝐵–1-1-onto→𝐵)
104	96, 102, 103	syl2anc 691	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑆‘𝐺):𝐵–1-1-onto→𝐵)
105		f1ococnv1 6078	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐺):𝐵–1-1-onto→𝐵 → (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) = ( I ↾ 𝐵))
106	104, 105	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) = ( I ↾ 𝐵))
107	106	coeq1d 5205	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → ((^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) ∘ 𝑧) = (( I ↾ 𝐵) ∘ 𝑧))
108	60	ad2antrl 760	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑧 ∈ 𝑇)
109	1, 6, 36	ltrn1o 34428	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → 𝑧:𝐵–1-1-onto→𝐵)
110	96, 108, 109	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑧:𝐵–1-1-onto→𝐵)
111		f1of 6050	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵–1-1-onto→𝐵 → 𝑧:𝐵⟶𝐵)
112		fcoi2 5992	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑧) = 𝑧)
113	110, 111, 112	3syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (( I ↾ 𝐵) ∘ 𝑧) = 𝑧)
114	107, 113	eqtr2d 2645	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑧 = ((^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) ∘ 𝑧))
115		simprrl 800	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 = (𝑥 ∘ 𝑧))
116	93	coeq1d 5205	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑥 ∘ 𝑧) = ((𝑆‘𝐺) ∘ 𝑧))
117	115, 116	eqtrd 2644	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 = ((𝑆‘𝐺) ∘ 𝑧))
118	117	coeq1d 5205	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝐹 ∘ ^◡(𝑆‘𝐺)) = (((𝑆‘𝐺) ∘ 𝑧) ∘ ^◡(𝑆‘𝐺)))
119	6, 36	ltrncnv 34450	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → ^◡(𝑆‘𝐺) ∈ 𝑇)
120	96, 102, 119	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → ^◡(𝑆‘𝐺) ∈ 𝑇)
121	6, 36	ltrnco 35025	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇)
122	96, 102, 108, 121	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇)
123	6, 36	ltrncom 35044	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ^◡(𝑆‘𝐺) ∈ 𝑇 ∧ ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇) → (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧)) = (((𝑆‘𝐺) ∘ 𝑧) ∘ ^◡(𝑆‘𝐺)))
124	96, 120, 122, 123	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧)) = (((𝑆‘𝐺) ∘ 𝑧) ∘ ^◡(𝑆‘𝐺)))
125	118, 124	eqtr4d 2647	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝐹 ∘ ^◡(𝑆‘𝐺)) = (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧)))
126	95, 114, 125	3eqtr4a 2670	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)))
127		simplrr 797	. . . . . . . . . . 11 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑤 = 𝑍)
128	127	adantl 481	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑤 = 𝑍)
129	126, 128	jca 553	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))
130	93, 94, 129	jca31 555	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)))
131	130	ex 449	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))))
132	131	pm4.71rd 665	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) ↔ (((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)))))
133	88, 132	bitrd 267	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩)) ↔ (((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)))))
134		simprrl 800	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 = (𝑥 ∘ 𝑧))
135		simpll1 1093	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
136	89	adantl 481	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 = (𝑦‘𝐺))
137	97	adantl 481	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑦 ∈ 𝐸)
138	135, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
139		simpl3l 1109	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
140	139	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
141	135, 138, 140, 55	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐺 ∈ 𝑇)
142	135, 137, 141, 57	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑦‘𝐺) ∈ 𝑇)
143	136, 142	eqeltrd 2688	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑥 ∈ 𝑇)
144	60	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑧 ∈ 𝑇)
145	6, 36	ltrnco 35025	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 ∘ 𝑧) ∈ 𝑇)
146	135, 143, 144, 145	syl3anc 1318	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑥 ∘ 𝑧) ∈ 𝑇)
147	134, 146	eqeltrd 2688	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐹 ∈ 𝑇)
148		simpl1l 1105	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝐾 ∈ HL)
149	148	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐾 ∈ HL)
150	149, 20	syl 17	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝐾 ∈ Lat)
151	1, 6, 36, 43	trlcl 34469	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ∈ 𝐵)
152	135, 144, 151	syl2anc 691	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑅‘𝑧) ∈ 𝐵)
153		simpl2l 1107	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝑋 ∈ 𝐵)
154	153	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑋 ∈ 𝐵)
155		simpl1r 1106	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝑊 ∈ 𝐻)
156	155	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑊 ∈ 𝐻)
157	156, 24	syl 17	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → 𝑊 ∈ 𝐵)
158	150, 154, 157, 26	syl3anc 1318	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑋 ∧ 𝑊) ∈ 𝐵)
159		simprlr 799	. . . . . . . . . . 11 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊))
160	159	ad2antrl 760	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊))
161	1, 2, 4	latmle1 16899	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑋)
162	150, 154, 157, 161	syl3anc 1318	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑋 ∧ 𝑊) ≤ 𝑋)
163	1, 2, 150, 152, 158, 154, 160, 162	lattrd 16881	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑅‘𝑧) ≤ 𝑋)
164	147, 137, 163	jca31 555	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋))
165		simprll 798	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝑥 = (𝑆‘𝐺))
166	165	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑥 = (𝑆‘𝐺))
167		simprlr 799	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝑦 = 𝑆)
168	167	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑦 = 𝑆)
169	168	fveq1d 6105	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑦‘𝐺) = (𝑆‘𝐺))
170	166, 169	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑥 = (𝑦‘𝐺))
171		simprlr 799	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑦 ∈ 𝐸)
172	170, 171	jca 553	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸))
173		simprrl 800	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)))
174	173	adantr 480	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)))
175		simpll1 1093	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
176		simprll 798	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐹 ∈ 𝑇)
177	168, 171	eqeltrrd 2689	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑆 ∈ 𝐸)
178	175, 52	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
179	139	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
180	175, 178, 179, 55	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐺 ∈ 𝑇)
181	175, 177, 180, 101	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑆‘𝐺) ∈ 𝑇)
182	175, 181, 119	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → ^◡(𝑆‘𝐺) ∈ 𝑇)
183	6, 36	ltrnco 35025	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡(𝑆‘𝐺) ∈ 𝑇) → (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇)
184	175, 176, 182, 183	syl3anc 1318	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇)
185	174, 184	eqeltrd 2688	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑧 ∈ 𝑇)
186		simprr 792	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑅‘𝑧) ≤ 𝑋)
187	2, 6, 36, 43	trlle 34489	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ≤ 𝑊)
188	175, 185, 187	syl2anc 691	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑅‘𝑧) ≤ 𝑊)
189	148	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐾 ∈ HL)
190	189, 20	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐾 ∈ Lat)
191	175, 185, 151	syl2anc 691	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑅‘𝑧) ∈ 𝐵)
192	153	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑋 ∈ 𝐵)
193	155	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑊 ∈ 𝐻)
194	193, 24	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑊 ∈ 𝐵)
195	1, 2, 4	latlem12 16901	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝑧) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝑅‘𝑧) ≤ 𝑋 ∧ (𝑅‘𝑧) ≤ 𝑊) ↔ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)))
196	190, 191, 192, 194, 195	syl13anc 1320	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (((𝑅‘𝑧) ≤ 𝑋 ∧ (𝑅‘𝑧) ≤ 𝑊) ↔ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)))
197	186, 188, 196	mpbi2and 958	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊))
198		simprrr 801	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → 𝑤 = 𝑍)
199	198	adantr 480	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑤 = 𝑍)
200	185, 197, 199	jca31 555	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍))
201	175, 181, 103	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑆‘𝐺):𝐵–1-1-onto→𝐵)
202	201, 105	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) = ( I ↾ 𝐵))
203	202	coeq2d 5206	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝐹 ∘ (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺))) = (𝐹 ∘ ( I ↾ 𝐵)))
204	1, 6, 36	ltrn1o 34428	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
205	175, 176, 204	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐹:𝐵–1-1-onto→𝐵)
206		f1of 6050	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
207		fcoi1 5991	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
208	205, 206, 207	3syl 18	. . . . . . . . . . . . . 14 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
209	203, 208	eqtr2d 2645	. . . . . . . . . . . . 13 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐹 = (𝐹 ∘ (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺))))
210		coass 5571	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)) = (𝐹 ∘ (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)))
211	209, 210	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐹 = ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)))
212	6, 36	ltrncom 35044	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇 ∧ (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇) → ((𝑆‘𝐺) ∘ (𝐹 ∘ ^◡(𝑆‘𝐺))) = ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)))
213	175, 181, 184, 212	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → ((𝑆‘𝐺) ∘ (𝐹 ∘ ^◡(𝑆‘𝐺))) = ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)))
214	211, 213	eqtr4d 2647	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐹 = ((𝑆‘𝐺) ∘ (𝐹 ∘ ^◡(𝑆‘𝐺))))
215	166, 174	coeq12d 5208	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑥 ∘ 𝑧) = ((𝑆‘𝐺) ∘ (𝐹 ∘ ^◡(𝑆‘𝐺))))
216	214, 215	eqtr4d 2647	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝐹 = (𝑥 ∘ 𝑧))
217	168	eqcomd 2616	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → 𝑆 = 𝑦)
218	216, 217	jca 553	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))
219	172, 200, 218	jca31 555	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)))
220	164, 219	impbida 873	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) ∧ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍))) → ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
221	220	pm5.32da 671	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) ↔ (((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋))))
222		df-3an 1033	. . . . . 6 ⊢ (((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) ↔ (((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
223	221, 222	syl6bbr 277	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍)) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) ↔ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋))))
224	48, 133, 223	3bitrd 293	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩)) ↔ ((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋))))
225	224	4exbidv 1841	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩)) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋))))
226		fvex 6113	. . . 4 ⊢ (𝑆‘𝐺) ∈ V
227	226	cnvex 7006	. . . . 5 ⊢ ^◡(𝑆‘𝐺) ∈ V
228	77, 227	coex 7011	. . . 4 ⊢ (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ V
229		fvex 6113	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
230	36, 229	eqeltri 2684	. . . . . 6 ⊢ 𝑇 ∈ V
231	230	mptex 6390	. . . . 5 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
232	44, 231	eqeltri 2684	. . . 4 ⊢ 𝑍 ∈ V
233		biidd 251	. . . 4 ⊢ (𝑥 = (𝑆‘𝐺) → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
234		eleq1 2676	. . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
235	234	anbi2d 736	. . . . 5 ⊢ (𝑦 = 𝑆 → ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ↔ (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸)))
236	235	anbi1d 737	. . . 4 ⊢ (𝑦 = 𝑆 → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
237		fveq2 6103	. . . . . 6 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (𝑅‘𝑧) = (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))))
238	237	breq1d 4593	. . . . 5 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → ((𝑅‘𝑧) ≤ 𝑋 ↔ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋))
239	238	anbi2d 736	. . . 4 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
240		biidd 251	. . . 4 ⊢ (𝑤 = 𝑍 → (((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
241	226, 78, 228, 232, 233, 236, 239, 240	ceqsex4v 3220	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = (𝑆‘𝐺) ∧ 𝑦 = 𝑆) ∧ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) ∧ 𝑤 = 𝑍) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋))
242	225, 241	syl6bb 275	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩)) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
243	13, 34, 242	3bitrd 293	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ⟨cop 4131 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 ^◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 ↦ cmpt2 6551 Basecbs 15695 +_gcplusg 15768 Scalarcsca 15771 lecple 15775 occoc 15776 joincjn 16767 meetcmee 16768 Latclat 16868 LSSumclsm 17872 LSubSpclss 18753 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 trLctrl 34463 TEndoctendo 35058 DVecHcdvh 35385 DIsoBcdib 35445 DIsoCcdic 35479 DIsoHcdih 35535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-0g 15925 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-cntz 17573 df-lsm 17874 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-lmod 18688 df-lss 18754 df-lsp 18793 df-lvec 18924 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring 35063 df-disoa 35336 df-dvech 35386 df-dib 35446 df-dic 35480 df-dih 35536

This theorem is referenced by: dihopelvalc 35556

W3C validator