dihopelvalcpre - Mathbox for Norm Megill

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Theorem dihopelvalcpre 41250

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 41238	. . 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 2827	. 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 1137	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3l 1202	. . . 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 41195	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐶‘𝑄) ∈ 𝑉)
18	14, 15, 17	syl2anc 584	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐶‘𝑄) ∈ 𝑉)
19		simp1l 1198	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
20	19	hllatd 39365	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21		simp2l 1200	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
22		simp1r 1199	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
23	1, 6	lhpbase 40000	. . . . . 6 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
24	22, 23	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐵)
25	1, 4	latmcl 18485	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
26	20, 21, 24, 25	syl3anc 1373	. . . 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 18510	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
28	20, 21, 24, 27	syl3anc 1373	. . . 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 41172	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉)
30	14, 26, 28, 29	syl12anc 837	. . 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 41119	. . 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 1373	. 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 3484	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		vex 3484	. . . . . . . . 9 ⊢ 𝑦 ∈ V
40	2, 5, 6, 34, 35, 36, 9, 37, 38, 39	dicopelval2 41183	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ↔ (𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸)))
41	14, 15, 40	syl2anc 584	. . . . . . 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 41150	. . . . . . . 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 837	. . . . . . 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 632	. . . . . 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 631	. . . . 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 1192	. . . . . . . . . . 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 779	. . . . . . . . . . . 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 780	. . . . . . . . . . . . 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 40021	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
52	48, 51	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))
53		simpl3l 1229	. . . . . . . . . . . . . 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 40581	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
55	48, 52, 53, 54	syl3anc 1373	. . . . . . . . . . . . 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 40769	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑦‘𝐺) ∈ 𝑇)
57	48, 50, 55, 56	syl3anc 1373	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 779	. . . . . . . . . . . 12 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → 𝑧 ∈ 𝑇)
60	59	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)
61		simprrr 782	. . . . . . . . . . . 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 40792	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑍 ∈ 𝐸)
63	48, 62	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)
64	61, 63	eqeltrd 2841	. . . . . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
66		eqid 2737	. . . . . . . . . . . 12 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
67	6, 35, 36, 10, 65, 31, 66	dvhopvadd 41095	. . . . . . . . . . 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 1381	. . . . . . . . . 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 41086	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+_g‘(Scalar‘𝑈)) = 𝑂)
71	48, 70	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)
72	71	oveqd 7448	. . . . . . . . . . 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 4880	. . . . . . . . . 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 2777	. . . . . . . . 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 2748	. . . . . . . 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 5481	. . . . . . . . 9 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
79	61	oveq2d 7447	. . . . . . . . . . . 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 40794	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸) → (𝑦𝑂𝑍) = 𝑦)
81	48, 50, 80	syl2anc 584	. . . . . . . . . . . 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 2777	. . . . . . . . . . 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 2748	. . . . . . . . . 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 630	. . . . . . . . 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 283	. . . . . . . 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 279	. . . . . . 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 579	. . . . . 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 775	. . . . . . . . . . 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 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))
90		simprrr 782	. . . . . . . . . . 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 6908	. . . . . . . . . 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 2780	. . . . . . . . 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 2743	. . . . . . . . 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 6285	. . . . . . . . . . 11 ⊢ ((^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) ∘ 𝑧) = (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧))
95		simpl1 1192	. . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑦 ∈ 𝐸)
97	96	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)
98	90, 97	eqeltrd 2841	. . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . 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 40769	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑆‘𝐺) ∈ 𝑇)
101	95, 98, 99, 100	syl3anc 1373	. . . . . . . . . . . . . . 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 40126	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → (𝑆‘𝐺):𝐵–1-1-onto→𝐵)
103	95, 101, 102	syl2anc 584	. . . . . . . . . . . . . 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 6877	. . . . . . . . . . . . . 14 ⊢ ((𝑆‘𝐺):𝐵–1-1-onto→𝐵 → (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) = ( I ↾ 𝐵))
105	103, 104	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))
106	105	coeq1d 5872	. . . . . . . . . . . 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 728	. . . . . . . . . . . . . 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 40126	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → 𝑧:𝐵–1-1-onto→𝐵)
109	95, 107, 108	syl2anc 584	. . . . . . . . . . . . 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 6848	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵–1-1-onto→𝐵 → 𝑧:𝐵⟶𝐵)
111		fcoi2 6783	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑧) = 𝑧)
112	109, 110, 111	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)
113	106, 112	eqtr2d 2778	. . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 5872	. . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . 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 5872	. . . . . . . . . . . 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 40148	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → ^◡(𝑆‘𝐺) ∈ 𝑇)
119	95, 101, 118	syl2anc 584	. . . . . . . . . . . . 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 40721	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇)
121	95, 101, 107, 120	syl3anc 1373	. . . . . . . . . . . . 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 40740	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 2803	. . . . . . . . . 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 778	. . . . . . . . . . 11 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑤 = 𝑍)
127	126	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)
128	125, 127	jca 511	. . . . . . . . 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 514	. . . . . . . 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 412	. . . . . . 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 562	. . . . . 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 279	. . . . 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 781	. . . . . . . . . 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 1213	. . . . . . . . . . 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 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))
136	96	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)
137	134, 51	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))
138		simpl3l 1229	. . . . . . . . . . . . . . 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 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))
140	134, 137, 139, 54	syl3anc 1373	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 728	. . . . . . . . . . 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 40721	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 ∘ 𝑧) ∈ 𝑇)
145	134, 142, 143, 144	syl3anc 1373	. . . . . . . . . 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 2841	. . . . . . . . 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 1225	. . . . . . . . . . . 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 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)
149	148	hllatd 39365	. . . . . . . . . 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 40166	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ∈ 𝐵)
151	134, 143, 150	syl2anc 584	. . . . . . . . . 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 1227	. . . . . . . . . . . 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 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)
154		simpl1r 1226	. . . . . . . . . . . . 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 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)
156	155, 23	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)
157	149, 153, 156, 25	syl3anc 1373	. . . . . . . . . 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 780	. . . . . . . . . . 11 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊))
159	158	ad2antrl 728	. . . . . . . . . 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 18509	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑋)
161	149, 153, 156, 160	syl3anc 1373	. . . . . . . . . 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 18491	. . . . . . . . 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 514	. . . . . . . 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 779	. . . . . . . . . . . 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 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))
166		simprlr 780	. . . . . . . . . . . . 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 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)
168	167	fveq1d 6908	. . . . . . . . . . 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 2780	. . . . . . . . . 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 780	. . . . . . . . . 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 511	. . . . . . . . 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 781	. . . . . . . . . . . 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 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)))
174		simpll1 1213	. . . . . . . . . . . 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 779	. . . . . . . . . . . 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 2842	. . . . . . . . . . . . . 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 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))
178	138	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))
179	174, 177, 178, 54	syl3anc 1373	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 40721	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡(𝑆‘𝐺) ∈ 𝑇) → (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇)
183	174, 175, 181, 182	syl3anc 1373	. . . . . . . . . . 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 2841	. . . . . . . . . 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 773	. . . . . . . . . . 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 40186	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ≤ 𝑊)
187	174, 184, 186	syl2anc 584	. . . . . . . . . . 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 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)
189	188	hllatd 39365	. . . . . . . . . . . 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 584	. . . . . . . . . . . 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 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)
192	154	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)
193	192, 23	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)
194	1, 2, 4	latlem12 18511	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 712	. . . . . . . . . 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 782	. . . . . . . . . . 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 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)
199	184, 196, 198	jca31 514	. . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 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))
202	201	coeq2d 5873	. . . . . . . . . . . . . 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 40126	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
204	174, 175, 203	syl2anc 584	. . . . . . . . . . . . . . 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 6848	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
206		fcoi1 6782	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
207	204, 205, 206	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)
208	202, 207	eqtr2d 2778	. . . . . . . . . . . . 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 6285	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)) = (𝐹 ∘ (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)))
210	208, 209	eqtr4di 2795	. . . . . . . . . . . 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 40740	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 2780	. . . . . . . . . . 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 5875	. . . . . . . . . . 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 2780	. . . . . . . . . 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 2743	. . . . . . . . . 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 511	. . . . . . . . 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 514	. . . . . . . 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 801	. . . . . . 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 579	. . . . . 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 1089	. . . . . 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 289	. . . . 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 305	. . . 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 1926	. . 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 6919	. . . 4 ⊢ (𝑆‘𝐺) ∈ V
226	225	cnvex 7947	. . . . 5 ⊢ ^◡(𝑆‘𝐺) ∈ V
227	76, 226	coex 7952	. . . 4 ⊢ (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ V
228	35	fvexi 6920	. . . . . 6 ⊢ 𝑇 ∈ V
229	228	mptex 7243	. . . . 5 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
230	43, 229	eqeltri 2837	. . . 4 ⊢ 𝑍 ∈ V
231		biidd 262	. . . 4 ⊢ (𝑥 = (𝑆‘𝐺) → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
232		eleq1 2829	. . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
233	232	anbi2d 630	. . . . 5 ⊢ (𝑦 = 𝑆 → ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ↔ (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸)))
234	233	anbi1d 631	. . . 4 ⊢ (𝑦 = 𝑆 → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
235		fveq2 6906	. . . . . 6 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (𝑅‘𝑧) = (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))))
236	235	breq1d 5153	. . . . 5 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → ((𝑅‘𝑧) ≤ 𝑋 ↔ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋))
237	236	anbi2d 630	. . . 4 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
238		biidd 262	. . . 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 3538	. . 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 287	. 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 305	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ⟨cop 4632 class class class wbr 5143 ↦ cmpt 5225 I cid 5577 ^◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ∈ cmpo 7433 Basecbs 17247 +_gcplusg 17297 Scalarcsca 17300 lecple 17304 occoc 17305 joincjn 18357 meetcmee 18358 Latclat 18476 LSSumclsm 19652 LSubSpclss 20929 Atomscatm 39264 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 trLctrl 40160 TEndoctendo 40754 DVecHcdvh 41080 DIsoBcdib 41140 DIsoCcdic 41174 DIsoHcdih 41230

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-riotaBAD 38954

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-undef 8298 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-dvr 20401 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tendo 40757 df-edring 40759 df-disoa 41031 df-dvech 41081 df-dib 41141 df-dic 41175 df-dih 41231

This theorem is referenced by: dihopelvalc 41251

W3C validator