dihopelvalcpre - Mathbox for Norm Megill

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Theorem dihopelvalcpre 41877

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 41865	. . 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 2850	. 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 1150	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp3l 1216	. . . 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 41822	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐶‘𝑄) ∈ 𝑉)
18	14, 15, 17	syl2anc 593	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → (𝐶‘𝑄) ∈ 𝑉)
19		simp1l 1212	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ HL)
20	19	hllatd 39993	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝐾 ∈ Lat)
21		simp2l 1214	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑋 ∈ 𝐵)
22		simp1r 1213	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → 𝑊 ∈ 𝐻)
23	1, 6	lhpbase 40627	. . . . . 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 18474	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
26	20, 21, 24, 25	syl3anc 1392	. . . 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 18499	. . . . 5 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
28	20, 21, 24, 27	syl3anc 1392	. . . 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 41799	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝑁‘(𝑋 ∧ 𝑊)) ∈ 𝑉)
30	14, 26, 28, 29	syl12anc 847	. . 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 41746	. . 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 1392	. 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 3460	. . . . . . . . 9 ⊢ 𝑥 ∈ V
39		vex 3460	. . . . . . . . 9 ⊢ 𝑦 ∈ V
40	2, 5, 6, 34, 35, 36, 9, 37, 38, 39	dicopelval2 41810	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐶‘𝑄) ↔ (𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸)))
41	14, 15, 40	syl2anc 593	. . . . . . 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 41777	. . . . . . . 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 847	. . . . . . 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 641	. . . . . 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 640	. . . . 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 1206	. . . . . . . . . . 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 788	. . . . . . . . . . . 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 789	. . . . . . . . . . . . 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 40648	. . . . . . . . . . . . . . 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 1243	. . . . . . . . . . . . . 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 41208	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
55	48, 52, 53, 54	syl3anc 1392	. . . . . . . . . . . . 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 41396	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑦‘𝐺) ∈ 𝑇)
57	48, 50, 55, 56	syl3anc 1392	. . . . . . . . . . . 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 2864	. . . . . . . . . . 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 788	. . . . . . . . . . . 12 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → 𝑧 ∈ 𝑇)
60	59	adantl 485	. . . . . . . . . . 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 791	. . . . . . . . . . . 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 41419	. . . . . . . . . . . . 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 2864	. . . . . . . . . . 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 2764	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
66		eqid 2764	. . . . . . . . . . . 12 ⊢ (+_g‘(Scalar‘𝑈)) = (+_g‘(Scalar‘𝑈))
67	6, 35, 36, 10, 65, 31, 66	dvhopvadd 41722	. . . . . . . . . . 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 1400	. . . . . . . . . 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 41713	. . . . . . . . . . . . 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 7415	. . . . . . . . . . 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 4840	. . . . . . . . . 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 2799	. . . . . . . . 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 2775	. . . . . . . 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 5446	. . . . . . . . 9 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑦𝑂𝑤)⟩ ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑦𝑂𝑤)))
79	61	oveq2d 7414	. . . . . . . . . . . 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 41421	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑦 ∈ 𝐸) → (𝑦𝑂𝑍) = 𝑦)
81	48, 50, 80	syl2anc 593	. . . . . . . . . . . 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 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))) → (𝑦𝑂𝑤) = 𝑦)
83	82	eqeq2d 2775	. . . . . . . . . 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 639	. . . . . . . . 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 285	. . . . . . . 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 281	. . . . . . 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 587	. . . . . 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 784	. . . . . . . . . . 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 485	. . . . . . . . . 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 791	. . . . . . . . . . 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 6871	. . . . . . . . . 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 2802	. . . . . . . . 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 2770	. . . . . . . . 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 6255	. . . . . . . . . . 11 ⊢ ((^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)) ∘ 𝑧) = (^◡(𝑆‘𝐺) ∘ ((𝑆‘𝐺) ∘ 𝑧))
95		simpl1 1206	. . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑦 ∈ 𝐸)
97	96	adantl 485	. . . . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . 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 41396	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) → (𝑆‘𝐺) ∈ 𝑇)
101	95, 98, 99, 100	syl3anc 1392	. . . . . . . . . . . . . . 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 40753	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → (𝑆‘𝐺):𝐵–1-1-onto→𝐵)
103	95, 101, 102	syl2anc 593	. . . . . . . . . . . . . 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 6838	. . . . . . . . . . . . . 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 5835	. . . . . . . . . . . 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 738	. . . . . . . . . . . . . 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 40753	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → 𝑧:𝐵–1-1-onto→𝐵)
109	95, 107, 108	syl2anc 593	. . . . . . . . . . . . 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 6808	. . . . . . . . . . . . 13 ⊢ (𝑧:𝐵–1-1-onto→𝐵 → 𝑧:𝐵⟶𝐵)
111		fcoi2 6741	. . . . . . . . . . . . 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 2800	. . . . . . . . . . 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 790	. . . . . . . . . . . . . 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 5835	. . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . 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 5835	. . . . . . . . . . . 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 40775	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇) → ^◡(𝑆‘𝐺) ∈ 𝑇)
119	95, 101, 118	syl2anc 593	. . . . . . . . . . . . 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 41348	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆‘𝐺) ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → ((𝑆‘𝐺) ∘ 𝑧) ∈ 𝑇)
121	95, 101, 107, 120	syl3anc 1392	. . . . . . . . . . . . 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 41367	. . . . . . . . . . . . 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 1392	. . . . . . . . . . . 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 2802	. . . . . . . . . . 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 2825	. . . . . . . . . 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 787	. . . . . . . . . . 11 ⊢ ((((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) ∧ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = 𝑦)) → 𝑤 = 𝑍)
127	126	adantl 485	. . . . . . . . . 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 519	. . . . . . . . 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 522	. . . . . . . 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 416	. . . . . . 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 570	. . . . . 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 281	. . . . 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 790	. . . . . . . . . 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 1227	. . . . . . . . . . 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 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))) → 𝑥 = (𝑦‘𝐺))
136	96	adantl 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))) ∧ (((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 1243	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . 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 1392	. . . . . . . . . . . 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 2864	. . . . . . . . . . 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 738	. . . . . . . . . . 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 41348	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇) → (𝑥 ∘ 𝑧) ∈ 𝑇)
145	134, 142, 143, 144	syl3anc 1392	. . . . . . . . . 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 2864	. . . . . . . . 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 1239	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 39993	. . . . . . . . . 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 40793	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ∈ 𝐵)
151	134, 143, 150	syl2anc 593	. . . . . . . . . 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 1241	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 1240	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 1392	. . . . . . . . . 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 789	. . . . . . . . . . 11 ⊢ (((𝑥 = (𝑦‘𝐺) ∧ 𝑦 ∈ 𝐸) ∧ ((𝑧 ∈ 𝑇 ∧ (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊)) ∧ 𝑤 = 𝑍)) → (𝑅‘𝑧) ≤ (𝑋 ∧ 𝑊))
159	158	ad2antrl 738	. . . . . . . . . 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 18498	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑋)
161	149, 153, 156, 160	syl3anc 1392	. . . . . . . . . 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 18480	. . . . . . . . 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 522	. . . . . . . 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 788	. . . . . . . . . . . 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 484	. . . . . . . . . . 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 789	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 6871	. . . . . . . . . . 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 2802	. . . . . . . . . 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 789	. . . . . . . . . 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 519	. . . . . . . . 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 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))) → 𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)))
173	172	adantr 484	. . . . . . . . . . 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 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))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
175		simprll 788	. . . . . . . . . . . 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 2865	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . . 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 1392	. . . . . . . . . . . . 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 593	. . . . . . . . . . . 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 41348	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ^◡(𝑆‘𝐺) ∈ 𝑇) → (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ 𝑇)
183	174, 175, 181, 182	syl3anc 1392	. . . . . . . . . . 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 2864	. . . . . . . . . 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 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))) ∧ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)) → (𝑅‘𝑧) ≤ 𝑋)
186	2, 6, 35, 42	trlle 40813	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑧 ∈ 𝑇) → (𝑅‘𝑧) ≤ 𝑊)
187	174, 184, 186	syl2anc 593	. . . . . . . . . . 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 484	. . . . . . . . . . . . 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 39993	. . . . . . . . . . . 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 593	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 484	. . . . . . . . . . . . 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 18500	. . . . . . . . . . . 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 1393	. . . . . . . . . . 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 722	. . . . . . . . . 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 791	. . . . . . . . . . 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 484	. . . . . . . . . 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 522	. . . . . . . . 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 593	. . . . . . . . . . . . . . . 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 5836	. . . . . . . . . . . . . 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 40753	. . . . . . . . . . . . . . . 16 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
204	174, 175, 203	syl2anc 593	. . . . . . . . . . . . . . 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 6808	. . . . . . . . . . . . . . 15 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
206		fcoi1 6740	. . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 6255	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∘ ^◡(𝑆‘𝐺)) ∘ (𝑆‘𝐺)) = (𝐹 ∘ (^◡(𝑆‘𝐺) ∘ (𝑆‘𝐺)))
210	208, 209	eqtr4di 2817	. . . . . . . . . . . 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 41367	. . . . . . . . . . . . 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 1392	. . . . . . . . . . . 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 2802	. . . . . . . . . . 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 5838	. . . . . . . . . . 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 2802	. . . . . . . . . 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 2770	. . . . . . . . . 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 519	. . . . . . . . 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 522	. . . . . . . 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 810	. . . . . . 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 587	. . . . . 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 1101	. . . . . 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 291	. . . . 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 307	. . . 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 1948	. . 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 6882	. . . 4 ⊢ (𝑆‘𝐺) ∈ V
226	225	cnvex 7908	. . . . 5 ⊢ ^◡(𝑆‘𝐺) ∈ V
227	76, 226	coex 7913	. . . 4 ⊢ (𝐹 ∘ ^◡(𝑆‘𝐺)) ∈ V
228	35	fvexi 6883	. . . . . 6 ⊢ 𝑇 ∈ V
229	228	mptex 7209	. . . . 5 ⊢ (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
230	43, 229	eqeltri 2860	. . . 4 ⊢ 𝑍 ∈ V
231		biidd 264	. . . 4 ⊢ (𝑥 = (𝑆‘𝐺) → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
232		eleq1 2852	. . . . . 6 ⊢ (𝑦 = 𝑆 → (𝑦 ∈ 𝐸 ↔ 𝑆 ∈ 𝐸))
233	232	anbi2d 639	. . . . 5 ⊢ (𝑦 = 𝑆 → ((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ↔ (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸)))
234	233	anbi1d 640	. . . 4 ⊢ (𝑦 = 𝑆 → (((𝐹 ∈ 𝑇 ∧ 𝑦 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋)))
235		fveq2 6869	. . . . . 6 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (𝑅‘𝑧) = (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))))
236	235	breq1d 5112	. . . . 5 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → ((𝑅‘𝑧) ≤ 𝑋 ↔ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋))
237	236	anbi2d 639	. . . 4 ⊢ (𝑧 = (𝐹 ∘ ^◡(𝑆‘𝐺)) → (((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘𝑧) ≤ 𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸) ∧ (𝑅‘(𝐹 ∘ ^◡(𝑆‘𝐺))) ≤ 𝑋)))
238		biidd 264	. . . 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 3509	. . 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 289	. 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 307	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 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∃wex 1801 ∈ wcel 2144 Vcvv 3456 ⟨cop 4590 class class class wbr 5102 ↦ cmpt 5183 I cid 5543 ^◡ccnv 5648 ↾ cres 5651 ∘ ccom 5653 ⟶wf 6519 –1-1-onto→wf1o 6522 ‘cfv 6523 ℩crio 7354 (class class class)co 7398 ∈ cmpo 7400 Basecbs 17247 +_gcplusg 17288 Scalarcsca 17291 lecple 17295 occoc 17296 joincjn 18345 meetcmee 18346 Latclat 18465 LSSumclsm 19676 LSubSpclss 21000 Atomscatm 39892 HLchlt 39979 LHypclh 40613 LTrncltrn 40730 trLctrl 40787 TEndoctendo 41381 DVecHcdvh 41707 DIsoBcdib 41767 DIsoCcdic 41801 DIsoHcdih 41857

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 39582

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-tpos 8208 df-undef 8255 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8680 df-map 8812 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-n0 12484 df-z 12571 df-uz 12842 df-fz 13515 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-sca 17304 df-vsca 17305 df-0g 17472 df-proset 18328 df-poset 18347 df-plt 18362 df-lub 18378 df-glb 18379 df-join 18380 df-meet 18381 df-p0 18457 df-p1 18458 df-lat 18466 df-clat 18533 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-grp 18980 df-minusg 18981 df-sbg 18982 df-subg 19167 df-cntz 19359 df-lsm 19678 df-cmn 19824 df-abl 19825 df-mgp 20189 df-rng 20201 df-ur 20234 df-ring 20287 df-oppr 20388 df-dvdsr 20408 df-unit 20409 df-invr 20439 df-dvr 20452 df-drng 20783 df-lmod 20931 df-lss 21001 df-lsp 21041 df-lvec 21172 df-oposet 39805 df-ol 39807 df-oml 39808 df-covers 39895 df-ats 39896 df-atl 39927 df-cvlat 39951 df-hlat 39980 df-llines 40127 df-lplanes 40128 df-lvols 40129 df-lines 40130 df-psubsp 40132 df-pmap 40133 df-padd 40425 df-lhyp 40617 df-laut 40618 df-ldil 40733 df-ltrn 40734 df-trl 40788 df-tendo 41384 df-edring 41386 df-disoa 41658 df-dvech 41708 df-dib 41768 df-dic 41802 df-dih 41858

This theorem is referenced by: dihopelvalc 41878

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