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Theorem tgaaddcpbllem1 29228

Description: Lemma for tgaaddcpbl 29231. (Contributed by Thierry Arnoux, 2-Aug-2026.)

**Hypotheses**
Ref	Expression
tgaaddcpbl.p	⊢ 𝑃 = (Base‘𝐺)
tgaaddcpbl.i	⊢ 𝐼 = (Itv‘𝐺)
tgaaddcpbl.l	⊢ 𝐿 = (LineG‘𝐺)
tgaaddcpbl.c	⊢ ∼ = (cgrA‘𝐺)
tgaaddcpbl.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}
tgaaddcpbl.q	⊢ 𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}
tgaaddcpbl.1	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgaaddcpbl.s	⊢ (𝜑 → 𝑆 ∈ 𝑃)
tgaaddcpbl.t	⊢ (𝜑 → 𝑇 ∈ 𝑃)
tgaaddcpbl.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
tgaaddcpbl.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
tgaaddcpbl.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
tgaaddcpbl.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
tgaaddcpbl.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
tgaaddcpbl.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
tgaaddcpbl.2	⊢ (𝜑 → 𝑌 ≠ 𝑆)
tgaaddcpbl.3	⊢ (𝜑 → 𝑉 ≠ 𝑇)
tgaaddcpbl.4	⊢ (𝜑 → 𝑋𝑂𝑍)
tgaaddcpbl.5	⊢ (𝜑 → 𝑈𝑄𝑊)
tgaaddcpbl.6	⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
tgaaddcpbl.7	⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
tgaaddcpbllem3.1	⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))
tgaaddcpbllem1.1	⊢ 𝐾 = (hlG‘𝐺)
tgaaddcpbllem1.2	⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
tgaaddcpbllem1.3	⊢ (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))
tgaaddcpbllem1.4	⊢ (𝜑 → 𝑅(𝐾‘𝑌)𝑆)

**Assertion**
Ref	Expression
tgaaddcpbllem1	⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

Distinct variable groups: ∼ ,𝑠 𝑡, ∼ 𝐺,𝑐,𝑑,𝑡 𝐺,𝑠 𝐼,𝑎,𝑏,𝑠 𝐼,𝑐,𝑑,𝑡 𝑡,𝐾 𝐿,𝑎,𝑏,𝑠 𝐿,𝑐,𝑑,𝑡 𝑂,𝑠 𝑃,𝑎,𝑏,𝑠 𝑃,𝑐,𝑑,𝑡 𝑄,𝑐,𝑑,𝑡 𝑡,𝑅 𝑆,𝑎,𝑏,𝑠 𝑡,𝑆 𝑇,𝑐,𝑑,𝑡 𝑈,𝑐,𝑑,𝑡 𝑈,𝑠 𝑉,𝑐,𝑑,𝑡 𝑉,𝑠 𝑊,𝑠 𝑡,𝑊 𝑋,𝑠 𝑡,𝑋 𝑌,𝑎,𝑏,𝑠 𝑡,𝑌 𝑍,𝑠 𝑡,𝑍 𝜑,𝑠 𝜑,𝑡 Allowed substitution hints: 𝜑(𝑎, 𝑏, 𝑐, 𝑑) 𝑄(𝑠, 𝑎, 𝑏) ∼ (𝑎, 𝑏, 𝑐, 𝑑) 𝑅(𝑠, 𝑎, 𝑏, 𝑐, 𝑑) 𝑆(𝑐, 𝑑) 𝑇(𝑠, 𝑎, 𝑏) 𝑈(𝑎, 𝑏) 𝐺(𝑎, 𝑏) 𝐾(𝑠, 𝑎, 𝑏, 𝑐, 𝑑) 𝑂(𝑡, 𝑎, 𝑏, 𝑐, 𝑑) 𝑉(𝑎, 𝑏) 𝑊(𝑎, 𝑏, 𝑐, 𝑑) 𝑋(𝑎, 𝑏, 𝑐, 𝑑) 𝑌(𝑐, 𝑑) 𝑍(𝑎, 𝑏, 𝑐, 𝑑)

Proof of Theorem tgaaddcpbllem1 Dummy variables 𝑟 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tgaaddcpbl.c	. . . . . . . . . 10 ⊢ ∼ = (cgrA‘𝐺)
2	1	eqcomi 2769	. . . . . . . . 9 ⊢ (cgrA‘𝐺) = ∼
3	2	a1i 11	. . . . . . . 8 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (cgrA‘𝐺) = ∼ )
4		tgaaddcpbl.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐺)
5		tgaaddcpbl.i	. . . . . . . . 9 ⊢ 𝐼 = (Itv‘𝐺)
6		tgaaddcpbllem1.1	. . . . . . . . 9 ⊢ 𝐾 = (hlG‘𝐺)
7		tgaaddcpbl.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8	7	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝐺 ∈ TarskiG)
9	8	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝐺 ∈ TarskiG)
10		tgaaddcpbl.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋 ∈ 𝑃)
12		tgaaddcpbl.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13	12	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ∈ 𝑃)
14		tgaaddcpbl.z	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
15	14	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑍 ∈ 𝑃)
16	15	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑍 ∈ 𝑃)
17		tgaaddcpbl.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
18	17	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑈 ∈ 𝑃)
19		tgaaddcpbl.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
20	19	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ∈ 𝑃)
21		simpllr 788	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤 ∈ 𝑃)
22		simp-6r 800	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑢 ∈ 𝑃)
23	22	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ∈ 𝑃)
24		tgaaddcpbl.l	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = (LineG‘𝐺)
25		tgaaddcpbl.s	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
26		tgaaddcpbl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ≠ 𝑆)
27	4, 5, 24, 7, 12, 25, 26	tglinerflx1 28980	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ (𝑌𝐿𝑆))
28		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (dist‘𝐺) = (dist‘𝐺)
29		tgaaddcpbl.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}
30	4, 5, 24, 7, 12, 25, 26	tgelrnln 28977	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
31		tgaaddcpbl.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋𝑂𝑍)
32	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne1 29096	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
33		nelne2 3053	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑋)
34	27, 32, 33	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
35	34	necomd 3010	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
36	35	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋 ≠ 𝑌)
37	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne2 29097	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
38		nelne2 3053	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑍)
39	27, 37, 38	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
40	39	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑍)
41		eqid 2760	. . . . . . . . . . . 12 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
42		simp-7r 802	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋))
43	42	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑋) = (𝑉(dist‘𝐺)𝑢))
44	4, 28, 5, 9, 13, 11, 20, 23, 43	tgcgrcomlr 28821	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑋(dist‘𝐺)𝑌) = (𝑢(dist‘𝐺)𝑉))
45	44	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢(dist‘𝐺)𝑉) = (𝑋(dist‘𝐺)𝑌))
46		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑟 ∈ 𝑃)
47	46	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ 𝑃)
48		tgaaddcpbllem1.2	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
49	4, 24, 5, 7, 30, 48	tglnpt 28891	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑅 ∈ 𝑃)
50	49	ad6antr 749	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑅 ∈ 𝑃)
51	50	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ∈ 𝑃)
52		simp-5r 798	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟(𝐾‘𝑉)𝑇)
53	9	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝐺 ∈ TarskiG)
54	11	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑋 ∈ 𝑃)
55	51	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅 ∈ 𝑃)
56	23	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑢 ∈ 𝑃)
57	47	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑟 ∈ 𝑃)
58	13	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑌 ∈ 𝑃)
59	20	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑉 ∈ 𝑃)
60		tgaaddcpbl.t	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
61	60	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑇 ∈ 𝑃)
62	61	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑇 ∈ 𝑃)
63	18	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑈 ∈ 𝑃)
64	25	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑆 ∈ 𝑃)
65	64	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑆 ∈ 𝑃)
66	1	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ∼ = (cgrA‘𝐺))
67		tgaaddcpbl.6	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
68	66, 67	breqdi 5118	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
69	68	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
70	4, 5, 9, 6, 11, 13, 64, 18, 20, 61, 69	cgracom 29208	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑈𝑉𝑇”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑆”⟩)
71	70	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → ⟨“𝑈𝑉𝑇”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑆”⟩)
72		tgaaddcpbllem1.4	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑅(𝐾‘𝑌)𝑆)
73	72	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅(𝐾‘𝑌)𝑆)
74	4, 5, 6, 53, 63, 59, 62, 54, 58, 65, 71, 55, 73	cgrahl2 29203	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → ⟨“𝑈𝑉𝑇”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑅”⟩)
75	4, 5, 53, 6, 63, 59, 62, 54, 58, 55, 74	cgracom 29208	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → ⟨“𝑋𝑌𝑅”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
76		simp-9r 806	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑢(𝐾‘𝑉)𝑈)
77	4, 5, 6, 53, 54, 58, 55, 63, 59, 62, 75, 56, 76	cgrahl1 29202	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → ⟨“𝑋𝑌𝑅”⟩(cgrA‘𝐺)⟨“𝑢𝑉𝑇”⟩)
78		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑟(𝐾‘𝑉)𝑇)
79	4, 5, 6, 53, 54, 58, 55, 56, 59, 62, 77, 57, 78	cgrahl2 29203	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → ⟨“𝑋𝑌𝑅”⟩(cgrA‘𝐺)⟨“𝑢𝑉𝑟”⟩)
80	4, 5, 6, 10, 10, 12, 7, 35	hlid 28954	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋(𝐾‘𝑌)𝑋)
81	80	ad9antr 755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋(𝐾‘𝑌)𝑋)
82	81	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑋(𝐾‘𝑌)𝑋)
83		tgaaddcpbllem3.1	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))
84		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑅 = 𝑌)
85		tgaaddcpbllem1.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))
86	85	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑅 ∈ (𝑋𝐼𝑍))
87	84, 86	eqeltrrd 2861	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑌 ∈ (𝑋𝐼𝑍))
88	83, 87	mtand 828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
89	88	neqned 2962	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
90	89	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
91	90	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑌 ≠ 𝑅)
92	91	necomd 3010	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅 ≠ 𝑌)
93	4, 5, 6, 55, 54, 58, 53, 92	hlid 28954	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅(𝐾‘𝑌)𝑅)
94	43	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑌(dist‘𝐺)𝑋) = (𝑉(dist‘𝐺)𝑢))
95		simp-4r 796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅))
96	95	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑅) = (𝑉(dist‘𝐺)𝑟))
97	96	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑌(dist‘𝐺)𝑅) = (𝑉(dist‘𝐺)𝑟))
98	4, 5, 6, 53, 54, 58, 55, 56, 59, 57, 79, 54, 28, 55, 82, 93, 94, 97	cgracgr 29204	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑋(dist‘𝐺)𝑅) = (𝑢(dist‘𝐺)𝑟))
99	4, 28, 5, 53, 54, 55, 56, 57, 98	tgcgrcomlr 28821	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑅(dist‘𝐺)𝑋) = (𝑟(dist‘𝐺)𝑢))
100	52, 99	mpdan 700	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑅(dist‘𝐺)𝑋) = (𝑟(dist‘𝐺)𝑢))
101	48	ad10antr 757	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅 ∈ (𝑌𝐿𝑆))
102	52, 101	mpdan 700	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ∈ (𝑌𝐿𝑆))
103	32	ad9antr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
104		nelne2 3053	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑅 ≠ 𝑋)
105	102, 103, 104	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ≠ 𝑋)
106	4, 28, 5, 9, 51, 11, 47, 23, 100, 105	tgcgrneq 28824	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ≠ 𝑢)
107	106	necomd 3010	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ≠ 𝑟)
108		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑢𝐼𝑤))
109	85	ad9antr 755	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ∈ (𝑋𝐼𝑍))
110	100	eqcomd 2766	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑟(dist‘𝐺)𝑢) = (𝑅(dist‘𝐺)𝑋))
111	4, 28, 5, 9, 47, 23, 51, 11, 110	tgcgrcomlr 28821	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢(dist‘𝐺)𝑟) = (𝑋(dist‘𝐺)𝑅))
112		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍))
113	4, 28, 5, 9, 20, 47, 13, 51, 95	tgcgrcomlr 28821	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑟(dist‘𝐺)𝑉) = (𝑅(dist‘𝐺)𝑌))
114	4, 28, 5, 9, 23, 47, 21, 11, 51, 16, 20, 13, 107, 108, 109, 111, 112, 45, 113	axtg5seg 28806	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑤(dist‘𝐺)𝑉) = (𝑍(dist‘𝐺)𝑌))
115	4, 28, 5, 9, 21, 20, 16, 13, 114	tgcgrcomlr 28821	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍))
116	4, 28, 5, 9, 23, 47, 21, 11, 51, 16, 108, 109, 111, 112	tgcgrextend 28826	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢(dist‘𝐺)𝑤) = (𝑋(dist‘𝐺)𝑍))
117	4, 28, 5, 9, 23, 21, 11, 16, 116	tgcgrcomlr 28821	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑤(dist‘𝐺)𝑢) = (𝑍(dist‘𝐺)𝑋))
118	4, 28, 41, 9, 23, 20, 21, 11, 13, 16, 45, 115, 117	trgcgr 28858	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑢𝑉𝑤”⟩(cgrG‘𝐺)⟨“𝑋𝑌𝑍”⟩)
119	4, 28, 5, 41, 9, 23, 20, 21, 11, 13, 16, 118	trgcgrcom 28870	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrG‘𝐺)⟨“𝑢𝑉𝑤”⟩)
120	4, 5, 9, 6, 11, 13, 16, 23, 20, 21, 36, 40, 119	cgrcgra 29207	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑢𝑉𝑤”⟩)
121	52, 76	mpdan 700	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢(𝐾‘𝑉)𝑈)
122	4, 5, 6, 23, 18, 20, 9, 121	hlcomd 28949	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑈(𝐾‘𝑉)𝑢)
123	4, 5, 6, 9, 11, 13, 16, 23, 20, 21, 120, 18, 122	cgrahl1 29202	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑤”⟩)
124		tgaaddcpbl.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
125	124	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑊 ∈ 𝑃)
126		eqid 2760	. . . . . . . . . . . 12 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
127		eqid 2760	. . . . . . . . . . . 12 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
128	4, 28, 5, 24, 126, 7, 60, 127, 17	mircl 29012	. . . . . . . . . . 11 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ 𝑃)
129	128	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ 𝑃)
130	7	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝐺 ∈ TarskiG)
131	12	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑌 ∈ 𝑃)
132	25	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ 𝑃)
133	14	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ 𝑃)
134	26	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑌 ≠ 𝑆)
135		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑌𝐿𝑍))
136	39	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑌 ≠ 𝑍)
137	4, 5, 24, 130, 131, 133, 136	tglinecom 28982	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
138	135, 137	eleqtrd 2862	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑍𝐿𝑌))
139	39	necomd 3010	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
140	139	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ≠ 𝑌)
141	4, 5, 24, 130, 131, 132, 133, 134, 138, 140	lnrot1 28970	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ (𝑌𝐿𝑆))
142	37, 141	mtand 828	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑆 ∈ (𝑌𝐿𝑍))
143	39	neneqd 2960	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑌 = 𝑍)
144	142, 143	jca 521	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ 𝑆 ∈ (𝑌𝐿𝑍) ∧ ¬ 𝑌 = 𝑍))
145		ioran 999	. . . . . . . . . . . 12 ⊢ (¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ (¬ 𝑆 ∈ (𝑌𝐿𝑍) ∧ ¬ 𝑌 = 𝑍))
146	144, 145	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
147	146	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
148		tgaaddcpbl.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄𝑊)
149		tgaaddcpbl.q	. . . . . . . . . . . . . . 15 ⊢ 𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}
150	4, 28, 5, 149, 17, 124	islnopp 29094	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈𝑄𝑊 ↔ ((¬ 𝑈 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑊 ∈ (𝑉𝐿𝑇)) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑈𝐼𝑊))))
151	148, 150	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((¬ 𝑈 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑊 ∈ (𝑉𝐿𝑇)) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑈𝐼𝑊)))
152	151	simplld 780	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑇))
153	7	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝐺 ∈ TarskiG)
154	60	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑇 ∈ 𝑃)
155	17	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑈 ∈ 𝑃)
156	4, 28, 5, 24, 126, 153, 154, 127, 155	mirmir 29013	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘(((pInvG‘𝐺)‘𝑇)‘𝑈)) = 𝑈)
157		tgaaddcpbl.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
158	4, 5, 24, 7, 19, 60, 157	tgelrnln 28977	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑉𝐿𝑇) ∈ ran 𝐿)
159	158	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (𝑉𝐿𝑇) ∈ ran 𝐿)
160	4, 5, 24, 7, 19, 60, 157	tglinerflx2 28981	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑇 ∈ (𝑉𝐿𝑇))
161	160	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑇 ∈ (𝑉𝐿𝑇))
162	128	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ 𝑃)
163	19	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑉 ∈ 𝑃)
164		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
165	4, 24, 5, 153, 163, 162, 154, 164	colcom 28900	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (𝑇 ∈ ((((pInvG‘𝐺)‘𝑇)‘𝑈)𝐿𝑉) ∨ (((pInvG‘𝐺)‘𝑇)‘𝑈) = 𝑉))
166	4, 24, 5, 153, 162, 163, 154, 165	colrot1 28901	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ((((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇) ∨ 𝑉 = 𝑇))
167	157	neneqd 2960	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑉 = 𝑇)
168	167	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ¬ 𝑉 = 𝑇)
169	166, 168	olcnd 891	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇))
170	4, 28, 5, 24, 126, 153, 127, 159, 161, 169	mirln 29027	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∈ (𝑉𝐿𝑇))
171	156, 170	eqeltrrd 2861	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑈 ∈ (𝑉𝐿𝑇))
172	152, 171	mtand 828	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
173	172	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
174		tgaaddcpbl.7	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
175	66, 174	breqdi 5118	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
176	175	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
177	52, 92	mpdan 700	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ≠ 𝑌)
178	177	necomd 3010	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑅)
179	4, 28, 5, 9, 13, 51, 20, 47, 96, 178	tgcgrneq 28824	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑟)
180	179	necomd 3010	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ≠ 𝑉)
181	115	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑍) = (𝑉(dist‘𝐺)𝑤))
182	4, 28, 5, 9, 13, 16, 20, 21, 181, 40	tgcgrneq 28824	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑤)
183	4, 28, 5, 9, 47, 21, 51, 16, 112	tgcgrcomlr 28821	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑤(dist‘𝐺)𝑟) = (𝑍(dist‘𝐺)𝑅))
184	4, 28, 41, 9, 47, 20, 21, 51, 13, 16, 113, 115, 183	trgcgr 28858	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑟𝑉𝑤”⟩(cgrG‘𝐺)⟨“𝑅𝑌𝑍”⟩)
185	4, 5, 9, 6, 47, 20, 21, 51, 13, 16, 180, 182, 184	cgrcgra 29207	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑟𝑉𝑤”⟩(cgrA‘𝐺)⟨“𝑅𝑌𝑍”⟩)
186	4, 5, 6, 49, 25, 12, 7, 72	hlcomd 28949	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆(𝐾‘𝑌)𝑅)
187	186	ad9antr 755	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑆(𝐾‘𝑌)𝑅)
188	4, 5, 6, 9, 47, 20, 21, 51, 13, 16, 185, 64, 187	cgrahl1 29202	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑟𝑉𝑤”⟩(cgrA‘𝐺)⟨“𝑆𝑌𝑍”⟩)
189	4, 5, 9, 6, 47, 20, 21, 64, 13, 16, 188	cgracom 29208	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑟𝑉𝑤”⟩)
190	4, 5, 6, 47, 61, 20, 9, 52	hlcomd 28949	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑇(𝐾‘𝑉)𝑟)
191	4, 5, 6, 9, 64, 13, 16, 47, 20, 21, 189, 61, 190	cgrahl1 29202	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑤”⟩)
192	4, 5, 24, 7, 19, 60, 157	tglinecom 28982	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
193	192	fveq2d 6882	. . . . . . . . . . . 12 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
194	17, 152	eldifd 3910	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ (𝑃 ∖ (𝑉𝐿𝑇)))
195	4, 5, 126, 127, 149, 7, 158, 160, 194, 24	oppmir 29111	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄(((pInvG‘𝐺)‘𝑇)‘𝑈))
196	4, 28, 5, 149, 24, 158, 7, 17, 128, 195	oppcom 29099	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
197	4, 28, 5, 149, 24, 158, 7, 17, 124, 148	oppcom 29099	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊𝑄𝑈)
198	4, 5, 24, 149, 7, 158, 124, 128, 17, 197	lnopp2hpgb 29120	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈 ↔ 𝑊((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈)))
199	196, 198	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈))
200	193, 199	breqdi 5118	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝑇𝐿𝑉))(((pInvG‘𝐺)‘𝑇)‘𝑈))
201	200	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑊((hpG‘𝐺)‘(𝑇𝐿𝑉))(((pInvG‘𝐺)‘𝑇)‘𝑈))
202	193	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
203	196	ad9antr 755	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
204	158	ad9antr 755	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉𝐿𝑇) ∈ ran 𝐿)
205	4, 5, 6, 47, 61, 20, 9, 24, 52	hlln 28952	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑇𝐿𝑉))
206	192	ad9antr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
207	205, 206	eleqtrrd 2863	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑉𝐿𝑇))
208		nelne2 3053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑅 ≠ 𝑍)
209	48, 37, 208	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ≠ 𝑍)
210	209	neneqd 2960	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑅 = 𝑍)
211	210	ad9antr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑅 = 𝑍)
212	9	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝐺 ∈ TarskiG)
213	47	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑟 ∈ 𝑃)
214	21	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ 𝑃)
215	51	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑅 ∈ 𝑃)
216	16	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑍 ∈ 𝑃)
217	112	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍))
218	116	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑋(dist‘𝐺)𝑍) = (𝑢(dist‘𝐺)𝑤))
219	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne3 29098	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑋 ≠ 𝑍)
220	219	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋 ≠ 𝑍)
221	4, 28, 5, 9, 11, 16, 23, 21, 218, 220	tgcgrneq 28824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ≠ 𝑤)
222	4, 5, 24, 9, 23, 21, 221	tgelrnln 28977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢𝐿𝑤) ∈ ran 𝐿)
223	222	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ∈ ran 𝐿)
224	204	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑉𝐿𝑇) ∈ ran 𝐿)
225	4, 5, 24, 9, 23, 21, 221	tglinerflx1 28980	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ∈ (𝑢𝐿𝑤))
226	9	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝐺 ∈ TarskiG)
227	20	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ 𝑃)
228	61	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ 𝑃)
229	18	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ 𝑃)
230	157	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑇)
231	23	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ 𝑃)
232	34	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑋)
233	4, 28, 5, 9, 13, 11, 20, 23, 43, 232	tgcgrneq 28824	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑢)
234	233	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ≠ 𝑉)
235	234	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ≠ 𝑉)
236		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ (𝑉𝐿𝑇))
237	4, 5, 24, 226, 231, 227, 228, 235, 236, 230	lnrot2 28971	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ (𝑢𝐿𝑉))
238	4, 5, 24, 7, 19, 60, 157	tglinerflx1 28980	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑉 ∈ (𝑉𝐿𝑇))
239		nelne2 3053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑉 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑈 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑈)
240	238, 152, 239	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
241	240	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
242	241	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ≠ 𝑉)
243	4, 5, 24, 226, 231, 227, 235	tgelrnln 28977	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑉) ∈ ran 𝐿)
244	4, 5, 6, 23, 18, 20, 9, 24, 121	hlln 28952	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ∈ (𝑈𝐿𝑉))
245	241	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑈 ≠ 𝑉)
246	4, 5, 24, 9, 18, 20, 245	tglinecom 28982	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑈𝐿𝑉) = (𝑉𝐿𝑈))
247	244, 246	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ∈ (𝑉𝐿𝑈))
248	240	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑈)
249	4, 5, 24, 9, 23, 20, 18, 234, 247, 248	lnrot2 28971	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑈 ∈ (𝑢𝐿𝑉))
250	249	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ (𝑢𝐿𝑉))
251	4, 5, 24, 226, 231, 227, 235	tglinerflx2 28981	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ (𝑢𝐿𝑉))
252	4, 5, 24, 226, 229, 227, 242, 242, 243, 250, 251	tglinethru 28983	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑉) = (𝑈𝐿𝑉))
253	237, 252	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ (𝑈𝐿𝑉))
254	4, 5, 24, 226, 227, 228, 229, 230, 253, 242	lnrot1 28970	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ (𝑉𝐿𝑇))
255	152	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → ¬ 𝑈 ∈ (𝑉𝐿𝑇))
256	254, 255	pm2.65da 829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑢 ∈ (𝑉𝐿𝑇))
257		nelne1 3052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 ∈ (𝑢𝐿𝑤) ∧ ¬ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
258	225, 256, 257	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
259	258	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
260	4, 5, 24, 9, 23, 21, 47, 221, 108	btwnlng1 28966	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑢𝐿𝑤))
261	260	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑟 ∈ (𝑢𝐿𝑤))
262	207	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑟 ∈ (𝑉𝐿𝑇))
263	261, 262	elind 4146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑟 ∈ ((𝑢𝐿𝑤) ∩ (𝑉𝐿𝑇)))
264	4, 5, 24, 9, 23, 21, 221	tglinerflx2 28981	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤 ∈ (𝑢𝐿𝑤))
265	264	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ (𝑢𝐿𝑤))
266		simpr 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ (𝑉𝐿𝑇))
267	265, 266	elind 4146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ ((𝑢𝐿𝑤) ∩ (𝑉𝐿𝑇)))
268	4, 5, 24, 212, 223, 224, 259, 263, 267	tglineineq 28990	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑟 = 𝑤)
269	4, 28, 5, 212, 213, 214, 215, 216, 217, 268	tgcgreq 28823	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑅 = 𝑍)
270	211, 269	mtand 828	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑤 ∈ (𝑉𝐿𝑇))
271	4, 28, 5, 9, 23, 47, 21, 108	tgbtwncom 28830	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑤𝐼𝑢))
272	4, 28, 5, 149, 21, 23, 207, 270, 256, 271	islnoppd 29095	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤𝑄𝑢)
273	4, 28, 5, 149, 24, 204, 9, 21, 23, 272	oppcom 29099	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢𝑄𝑤)
274	238	ad9antr 755	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ∈ (𝑉𝐿𝑇))
275	4, 28, 5, 149, 24, 204, 9, 6, 23, 18, 21, 273, 274, 121	opphl 29109	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑈𝑄𝑤)
276	4, 28, 5, 149, 24, 204, 9, 18, 21, 275	oppcom 29099	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤𝑄𝑈)
277	4, 5, 24, 149, 9, 204, 21, 129, 18, 276	lnopp2hpgb 29120	. . . . . . . . . . . 12 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ((((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈 ↔ 𝑤((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈)))
278	203, 277	mpbid 235	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈))
279	202, 278	breqdi 5118	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤((hpG‘𝐺)‘(𝑇𝐿𝑉))(((pInvG‘𝐺)‘𝑇)‘𝑈))
280	4, 5, 28, 9, 64, 13, 16, 61, 20, 129, 24, 147, 173, 125, 21, 6, 176, 191, 201, 279	acopyeu 29221	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑊(𝐾‘𝑉)𝑤)
281	4, 5, 6, 9, 11, 13, 16, 18, 20, 21, 123, 125, 280	cgrahl2 29203	. . . . . . . 8 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑊”⟩)
282	3, 281	breqdi 5118	. . . . . . 7 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
283	282	anasss 472	. . . . . 6 ⊢ (((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ (𝑟 ∈ (𝑢𝐼𝑤) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
284	4, 28, 5, 8, 22, 46, 50, 15	axtgsegcon 28805	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → ∃𝑤 ∈ 𝑃 (𝑟 ∈ (𝑢𝐼𝑤) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)))
285	283, 284	r19.29a 3170	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
286	285	anasss 472	. . . 4 ⊢ ((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ (𝑟(𝐾‘𝑉)𝑇 ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
287	157	necomd 3010	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
288	4, 5, 6, 19, 12, 49, 7, 60, 28, 287, 90	hlcgrex 28961	. . . . 5 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟(𝐾‘𝑉)𝑇 ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)))
289	288	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ∃𝑟 ∈ 𝑃 (𝑟(𝐾‘𝑉)𝑇 ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)))
290	286, 289	r19.29a 3170	. . 3 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
291	290	anasss 472	. 2 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ (𝑢(𝐾‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
292	4, 5, 6, 19, 12, 10, 7, 17, 28, 241, 34	hlcgrex 28961	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 (𝑢(𝐾‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)))
293	291, 292	r19.29a 3170	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 ∖ cdif 3896 class class class wbr 5103 {copab 5167 ran crn 5656 ‘cfv 6533 (class class class)co 7413 ⟨“cs3 14913 Basecbs 17301 distcds 17351 TarskiGcstrkg 28768 Itvcitv 28774 LineGclng 28775 cgrGccgrg 28852 hlGchlg 28942 pInvGcmir 29003 hpGchpg 29114 cgrAccgra 29193

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 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-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12888 df-fz 13562 df-fzo 13710 df-hash 14395 df-word 14579 df-concat 14636 df-s1 14663 df-s2 14919 df-s3 14920 df-trkgc 28789 df-trkgb 28790 df-trkgcb 28791 df-trkgld 28793 df-trkg 28794 df-cgrg 28853 df-leg 28925 df-hlg 28943 df-mir 29004 df-rag 29048 df-perpg 29050 df-hpg 29115 df-mid 29158 df-lmi 29159 df-cgra 29194

This theorem is used by: tgaaddcpbllem2 29229 tgaaddcpbllem3 29230

W3C validator