dihopelvalcpre - Mathbox for Norm Megill

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Theorem dihopelvalcpre 42050

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 42038	. . 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 2849	. 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 1154	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3l 1220	. . . 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 41995	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐶‘𝑄) ∈ 𝑉)
18	14, 15, 17	syl2anc 595	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐶‘𝑄) ∈ 𝑉)
19		simp1l 1216	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
20	19	hllatd 40166	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21		simp2l 1218	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
22		simp1r 1217	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
23	1, 6	lhpbase 40800	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
24	22, 23	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵)
25	1, 4	latmcl 18500	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
26	20, 21, 24, 25	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
27	1, 2, 4	latmle2 18525	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
28	20, 21, 24, 27	syl3anc 1398	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑋 ∧ 𝑊) ≤ 𝑊)
29	1, 2, 6, 10, 8, 16	diblss 41972	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉)
30	14, 26, 28, 29	syl12anc 849	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉)
31		dihopelvalcp.d	. . . 4 ⊢ + = (+_g‘𝑈)
32	6, 10, 31, 16, 11	dvhopellsm 41919	. . 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⟩))))
33	14, 18, 30, 32	syl3anc 1398	. 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⟩))))
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 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
38		vex 3459	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		vex 3459	. . . . . . . . 9 ⊢ 𝑦 ∈ V
40	2, 5, 6, 34, 35, 36, 9, 37, 38, 39	dicopelval2 41983	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ↔ (𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸)))
41	14, 15, 40	syl2anc 595	. . . . . . 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)))
42		dihopelvalcp.r	. . . . . . . . 9 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
43		dihopelvalcp.z	. . . . . . . . 9 ⊢ 𝑍 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
44	1, 2, 6, 35, 42, 43, 8	dibopelval3 41950	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝑁‘(𝑋 ∧ 𝑊)) ↔ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)))
45	14, 26, 28, 44	syl12anc 849	. . . . . . 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)))
46	41, 45	anbi12d 643	. . . . . 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))))
47	46	anbi1d 642	. . . . 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⟩))))
48		simpl1 1210	. . . . . . . . . . 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))
49		simprll 790	. . . . . . . . . . . 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))
50		simprlr 791	. . . . . . . . . . . . 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)
51	2, 5, 6, 34	lhpocnel2 40821	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
52	48, 51	syl 18	. . . . . . . . . . . . . 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))
53		simpl3l 1247	. . . . . . . . . . . . . 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))
54	2, 5, 6, 35, 37	ltrniotacl 41381	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
55	48, 52, 53, 54	syl3anc 1398	. . . . . . . . . . . . 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)
56	6, 35, 36	tendocl 41569	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑦‘𝐺) ∈ 𝑇)
57	48, 50, 55, 56	syl3anc 1398	. . . . . . . . . . . 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)
58	49, 57	eqeltrd 2863	. . . . . . . . . . 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)
59		simprll 790	. . . . . . . . . . . 12 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → 𝑧 ∈ 𝑇)
60	59	adantl 486	. . . . . . . . . . 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)
61		simprrr 793	. . . . . . . . . . . 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)
62	1, 6, 35, 36, 43	tendo0cl 41592	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑍 ∈ 𝐸)
63	48, 62	syl 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))) → 𝑍 ∈ 𝐸)
64	61, 63	eqeltrd 2863	. . . . . . . . . . 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)
65		eqid 2763	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
66		eqid 2763	. . . . . . . . . . . 12 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
67	6, 35, 36, 10, 65, 31, 66	dvhopvadd 41895	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑧 ∈ 𝑇 ∧ 𝑤 ∈ 𝐸)) → (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩) = ⟨(𝑥 ∘ 𝑧), (𝑦(+_g‘(Scalar‘𝑈))𝑤)⟩)
68	48, 58, 50, 60, 64, 67	syl122anc 1406	. . . . . . . . . 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)⟩)
69		dihopelvalcp.o	. . . . . . . . . . . . . 14 ⊢ 𝑂 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑎‘ℎ) ∘ (𝑏‘ℎ))))
70	6, 35, 36, 10, 65, 69, 66	dvhfplusr 41886	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘(Scalar‘𝑈)) = 𝑂)
71	48, 70	syl 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))) → (+_g‘(Scalar‘𝑈)) = 𝑂)
72	71	oveqd 7427	. . . . . . . . . . 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))
73	72	opeq2d 4845	. . . . . . . . . 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)⟩)
74	68, 73	eqtrd 2798	. . . . . . . . 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)⟩)
75	74	eqeq2d 2774	. . . . . . . 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)⟩))
76		dihopelvalcp.f	. . . . . . . . . 10 ⊢ 𝐹 ∈ V
77		dihopelvalcp.s	. . . . . . . . . 10 ⊢ 𝑆 ∈ V
78	76, 77	opth 5458	. . . . . . . . 9 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
79	61	oveq2d 7426	. . . . . . . . . . . 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))
80	1, 6, 35, 36, 43, 69	tendo0plr 41594	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸) → (𝑦𝑂𝑍) = 𝑦)
81	48, 50, 80	syl2anc 595	. . . . . . . . . . . 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)
82	79, 81	eqtrd 2798	. . . . . . . . . . 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)
83	82	eqeq2d 2774	. . . . . . . . . 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))
84	83	anbi2d 641	. . . . . . . . 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)))
85	78, 84	bitrid 286	. . . . . . . 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)))
86	75, 85	bitrd 282	. . . . . . 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)))
87	86	pm5.32da 589	. . . . . 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))))
88		simplll 786	. . . . . . . . . . 11 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑥 = (𝑦‘𝐺))
89	88	adantl 486	. . . . . . . . . 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))
90		simprrr 793	. . . . . . . . . . 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)
91	90	fveq1d 6883	. . . . . . . . . 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))
92	89, 91	eqtr4d 2801	. . . . . . . . 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))
93	90	eqcomd 2769	. . . . . . . . 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)
94		coass 6267	. . . . . . . . . . 11 ⊢ ((^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) ∘ 𝑧) = (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧))
95		simpl1 1210	. . . . . . . . . . . . . . 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))
96		simpllr 787	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑦 ∈ 𝐸)
97	96	adantl 486	. . . . . . . . . . . . . . . . 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)
98	90, 97	eqeltrd 2863	. . . . . . . . . . . . . . . 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)
99	55	adantrr 729	. . . . . . . . . . . . . . . 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)
100	6, 35, 36	tendocl 41569	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑆‘𝐺) ∈ 𝑇)
101	95, 98, 99, 100	syl3anc 1398	. . . . . . . . . . . . . . 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)
102	1, 6, 35	ltrn1o 40926	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → (𝑆‘𝐺):𝐵–1-1-onto→𝐵)
103	95, 101, 102	syl2anc 595	. . . . . . . . . . . . . 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)
104		f1ococnv1 6850	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐺):𝐵–1-1-onto→𝐵 → (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) = ( I ↾ 𝐵))
105	103, 104	syl 18	. . . . . . . . . . . . 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))
106	105	coeq1d 5847	. . . . . . . . . . . 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))
107	59	ad2antrl 740	. . . . . . . . . . . . . 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)
108	1, 6, 35	ltrn1o 40926	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → 𝑧:𝐵–1-1-onto→𝐵)
109	95, 107, 108	syl2anc 595	. . . . . . . . . . . . 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)
110		f1of 6820	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵–1-1-onto→𝐵 → 𝑧:𝐵⟶𝐵)
111		fcoi2 6753	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑧) = 𝑧)
112	109, 110, 111	3syl 19	. . . . . . . . . . . 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)
113	106, 112	eqtr2d 2799	. . . . . . . . . . 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))
114		simprrl 792	. . . . . . . . . . . . . 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))
115	92	coeq1d 5847	. . . . . . . . . . . . . 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))
116	114, 115	eqtrd 2798	. . . . . . . . . . . . 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))
117	116	coeq1d 5847	. . . . . . . . . . . 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)))
118	6, 35	ltrncnv 40948	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → ^◡(𝑆‘𝐺) ∈ 𝑇)
119	95, 101, 118	syl2anc 595	. . . . . . . . . . . . 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)
120	6, 35	ltrnco 41521	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇)
121	95, 101, 107, 120	syl3anc 1398	. . . . . . . . . . . . 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)
122	6, 35	ltrncom 41540	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ^◡(𝑆‘𝐺) ∈ 𝑇 ∧ ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇) → (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧)) = (((𝑆‘𝐺) ∘ 𝑧) ∘ ^◡(𝑆‘𝐺)))
123	95, 119, 121, 122	syl3anc 1398	. . . . . . . . . . . 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)))
124	117, 123	eqtr4d 2801	. . . . . . . . . . 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)))
125	94, 113, 124	3eqtr4a 2824	. . . . . . . . . 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)))
126		simplrr 789	. . . . . . . . . . 11 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑤 = 𝑍)
127	126	adantl 486	. . . . . . . . . 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)
128	125, 127	jca 520	. . . . . . . . 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))
129	92, 93, 128	jca31 523	. . . . . . . 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)))
130	129	ex 417	. . . . . . 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))))
131	130	pm4.71rd 571	. . . . . 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)))))
132	87, 131	bitrd 282	. . . . 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)))))
133		simprrl 792	. . . . . . . . . 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))
134		simpll1 1231	. . . . . . . . . . 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))
135	88	adantl 486	. . . . . . . . . . . 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))
136	96	adantl 486	. . . . . . . . . . . . 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)
137	134, 51	syl 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))) ∧ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
138		simpl3l 1247	. . . . . . . . . . . . . . 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))
139	138	adantr 485	. . . . . . . . . . . . . 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))
140	134, 137, 139, 54	syl3anc 1398	. . . . . . . . . . . . 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)
141	134, 136, 140, 56	syl3anc 1398	. . . . . . . . . . . 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)
142	135, 141	eqeltrd 2863	. . . . . . . . . . 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)
143	59	ad2antrl 740	. . . . . . . . . . 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)
144	6, 35	ltrnco 41521	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 ∘ 𝑧) ∈ 𝑇)
145	134, 142, 143, 144	syl3anc 1398	. . . . . . . . . 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)
146	133, 145	eqeltrd 2863	. . . . . . . . 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)
147		simpl1l 1243	. . . . . . . . . . . 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)
148	147	adantr 485	. . . . . . . . . . 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)
149	148	hllatd 40166	. . . . . . . . . 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)
150	1, 6, 35, 42	trlcl 40966	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ∈ 𝐵)
151	134, 143, 150	syl2anc 595	. . . . . . . . . 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)
152		simpl2l 1245	. . . . . . . . . . . 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)
153	152	adantr 485	. . . . . . . . . . 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)
154		simpl1r 1244	. . . . . . . . . . . . 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)
155	154	adantr 485	. . . . . . . . . . . 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)
156	155, 23	syl 18	. . . . . . . . . . 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)
157	149, 153, 156, 25	syl3anc 1398	. . . . . . . . . 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)
158		simprlr 791	. . . . . . . . . . 11 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊))
159	158	ad2antrl 740	. . . . . . . . . 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))
160	1, 2, 4	latmle1 18524	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑋)
161	149, 153, 156, 160	syl3anc 1398	. . . . . . . . . 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)
162	1, 2, 149, 151, 157, 153, 159, 161	lattrd 18506	. . . . . . . . 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)
163	146, 136, 162	jca31 523	. . . . . . . 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))
164		simprll 790	. . . . . . . . . . . 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))
165	164	adantr 485	. . . . . . . . . . 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))
166		simprlr 791	. . . . . . . . . . . . 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)
167	166	adantr 485	. . . . . . . . . . . 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)
168	167	fveq1d 6883	. . . . . . . . . . 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))
169	165, 168	eqtr4d 2801	. . . . . . . . . 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))
170		simprlr 791	. . . . . . . . . 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)
171	169, 170	jca 520	. . . . . . . . 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))
172		simprrl 792	. . . . . . . . . . . 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)))
173	172	adantr 485	. . . . . . . . . . 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)))
174		simpll1 1231	. . . . . . . . . . . 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))
175		simprll 790	. . . . . . . . . . . 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)
176	167, 170	eqeltrrd 2864	. . . . . . . . . . . . . 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)
177	174, 51	syl 18	. . . . . . . . . . . . . . 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))
178	138	adantr 485	. . . . . . . . . . . . . . 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))
179	174, 177, 178, 54	syl3anc 1398	. . . . . . . . . . . . . 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)
180	174, 176, 179, 100	syl3anc 1398	. . . . . . . . . . . . 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)
181	174, 180, 118	syl2anc 595	. . . . . . . . . . . 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)
182	6, 35	ltrnco 41521	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡(𝑆‘𝐺) ∈ 𝑇) → (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇)
183	174, 175, 181, 182	syl3anc 1398	. . . . . . . . . . 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)
184	173, 183	eqeltrd 2863	. . . . . . . . . 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)
185		simprr 784	. . . . . . . . . . 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)
186	2, 6, 35, 42	trlle 40986	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ≤ 𝑊)
187	174, 184, 186	syl2anc 595	. . . . . . . . . . 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)
188	147	adantr 485	. . . . . . . . . . . . 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)
189	188	hllatd 40166	. . . . . . . . . . . 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)
190	174, 184, 150	syl2anc 595	. . . . . . . . . . . 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)
191	152	adantr 485	. . . . . . . . . . . 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)
192	154	adantr 485	. . . . . . . . . . . . 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)
193	192, 23	syl 18	. . . . . . . . . . . 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)
194	1, 2, 4	latlem12 18526	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ Lat ∧ ((𝑅‘𝑧) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (((𝑅‘𝑧) ≤ 𝑋 ∧ (𝑅‘𝑧) ≤ 𝑊) ↔ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)))
195	189, 190, 191, 193, 194	syl13anc 1399	. . . . . . . . . . 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)))
196	185, 187, 195	mpbi2and 724	. . . . . . . . . 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))
197		simprrr 793	. . . . . . . . . . 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)
198	197	adantr 485	. . . . . . . . . 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)
199	184, 196, 198	jca31 523	. . . . . . . . 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))
200	174, 180, 102	syl2anc 595	. . . . . . . . . . . . . . . 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)
201	200, 104	syl 18	. . . . . . . . . . . . . . 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))
202	201	coeq2d 5848	. . . . . . . . . . . . . 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)))
203	1, 6, 35	ltrn1o 40926	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
204	174, 175, 203	syl2anc 595	. . . . . . . . . . . . . . 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)
205		f1of 6820	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
206		fcoi1 6752	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
207	204, 205, 206	3syl 19	. . . . . . . . . . . . . 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)
208	202, 207	eqtr2d 2799	. . . . . . . . . . . . 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))))
209		coass 6267	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)) = (𝐹 ∘ (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)))
210	208, 209	eqtr4di 2816	. . . . . . . . . . . 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)))
211	6, 35	ltrncom 41540	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇 ∧ (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇) → ((𝑆‘𝐺) ∘ (𝐹 ∘ ^◡(𝑆‘𝐺))) = ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)))
212	174, 180, 183, 211	syl3anc 1398	. . . . . . . . . . . 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)))
213	210, 212	eqtr4d 2801	. . . . . . . . . . 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))))
214	165, 173	coeq12d 5850	. . . . . . . . . . 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))))
215	213, 214	eqtr4d 2801	. . . . . . . . . 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))
216	167	eqcomd 2769	. . . . . . . . . 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)
217	215, 216	jca 520	. . . . . . . . 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))
218	171, 199, 217	jca31 523	. . . . . . . 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)))
219	163, 218	impbida 812	. . . . . . 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)))
220	219	pm5.32da 589	. . . . . 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))))
221		df-3an 1105	. . . . . 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)))
222	220, 221	bitr4di 292	. . . . 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))))
223	47, 132, 222	3bitrd 308	. . . 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))))
224	223	4exbidv 1956	. . 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))))
225		fvex 6894	. . . 4 ⊢ (𝑆‘𝐺) ∈ V
226	225	cnvex 7918	. . . . 5 ⊢ ^◡(𝑆‘𝐺) ∈ V
227	76, 226	coex 7923	. . . 4 ⊢ (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ V
228	35	fvexi 6895	. . . . . 6 ⊢ 𝑇 ∈ V
229	228	mptex 7221	. . . . 5 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
230	43, 229	eqeltri 2859	. . . 4 ⊢ 𝑍 ∈ V
231		biidd 265	. . . 4 ⊢ (𝑥 = (𝑆‘𝐺) → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
232		eleq1 2851	. . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
233	232	anbi2d 641	. . . . 5 ⊢ (𝑦 = 𝑆 → ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ↔ (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸)))
234	233	anbi1d 642	. . . 4 ⊢ (𝑦 = 𝑆 → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
235		fveq2 6881	. . . . . 6 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (𝑅‘𝑧) = (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))))
236	235	breq1d 5119	. . . . 5 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → ((𝑅‘𝑧) ≤ 𝑋 ↔ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋))
237	236	anbi2d 641	. . . 4 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
238		biidd 265	. . . 4 ⊢ (𝑤 = 𝑍 → (((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
239	225, 77, 227, 230, 231, 234, 237, 238	ceqsex4v 3508	. . 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))
240	224, 239	bitrdi 290	. 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)))
241	13, 33, 240	3bitrd 308	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 Vcvv 3455 ⟨cop 4595 class class class wbr 5109 ↦ cmpt 5192 I cid 5555 ^◡ccnv 5660 ↾ cres 5663 ∘ ccom 5665 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 ∈ cmpo 7412 Basecbs 17273 +_gcplusg 17314 Scalarcsca 17317 lecple 17321 occoc 17322 joincjn 18371 meetcmee 18372 Latclat 18491 LSSumclsm 19708 LSubSpclss 21061 Atomscatm 40065 HLchlt 40152 LHypclh 40786 LTrncltrn 40903 trLctrl 40960 TEndoctendo 41554 DVecHcdvh 41880 DIsoBcdib 41940 DIsoCcdic 41974 DIsoHcdih 42030

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-riotaBAD 39755

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-sca 17330 df-vsca 17331 df-0g 17498 df-proset 18354 df-poset 18373 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18492 df-clat 18559 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-cntz 19391 df-lsm 19710 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-dvr 20488 df-drng 20838 df-lmod 20992 df-lss 21062 df-lsp 21102 df-lvec 21233 df-oposet 39978 df-ol 39980 df-oml 39981 df-covers 40068 df-ats 40069 df-atl 40100 df-cvlat 40124 df-hlat 40153 df-llines 40300 df-lplanes 40301 df-lvols 40302 df-lines 40303 df-psubsp 40305 df-pmap 40306 df-padd 40598 df-lhyp 40790 df-laut 40791 df-ldil 40906 df-ltrn 40907 df-trl 40961 df-tendo 41557 df-edring 41559 df-disoa 41831 df-dvech 41881 df-dib 41941 df-dic 41975 df-dih 42031

This theorem is used by: dihopelvalc 42051

W3C validator