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Theorem tgaaddcpbllem1 29342

Description: Lemma for tgaaddcpbl 29345. (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 2770	. . . . . . . . 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 29094	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ (𝑌𝐿𝑆))
28		eqid 2761	. . . . . . . . . . . . . . 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 29091	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
31		tgaaddcpbl.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋𝑂𝑍)
32	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne1 29210	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
33		nelne2 3054	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑋)
34	27, 32, 33	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
35	34	necomd 3011	. . . . . . . . . . . 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 29211	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
38		nelne2 3054	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . 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 29005	. . . . . . . . . . . . . . . . 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 29322	. . . . . . . . . . . . . . . . . . . . . . . . 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 29317	. . . . . . . . . . . . . . . . . . . . . . 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 29322	. . . . . . . . . . . . . . . . . . . . . 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 29316	. . . . . . . . . . . . . . . . . . . . 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 29317	. . . . . . . . . . . . . . . . . . . 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 29068	. . . . . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑌 ∈ (𝑋𝐼𝑍))
88	83, 87	mtand 828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
89	88	neqned 2963	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
90	89	necomd 3011	. . . . . . . . . . . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . . . . . . . . . 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 29068	. . . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . 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 29318	. . . . . . . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . . . . . . 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 3054	. . . . . . . . . . . . . . . . . 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 28938	. . . . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . . . 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 28920	. . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . 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 28940	. . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . 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 28972	. . . . . . . . . . . 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 28984	. . . . . . . . . . 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 29321	. . . . . . . . . 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 29063	. . . . . . . . . 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 29316	. . . . . . . . 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 2761	. . . . . . . . . . . 12 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
127		eqid 2761	. . . . . . . . . . . 12 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
128	4, 28, 5, 24, 126, 7, 60, 127, 17	mircl 29126	. . . . . . . . . . 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 29096	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
138	135, 137	eleqtrd 2863	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑍𝐿𝑌))
139	39	necomd 3011	. . . . . . . . . . . . . . . 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 29084	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ (𝑌𝐿𝑆))
142	37, 141	mtand 828	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑆 ∈ (𝑌𝐿𝑍))
143	39	neneqd 2961	. . . . . . . . . . . . 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 29208	. . . . . . . . . . . . . 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 29127	. . . . . . . . . . . . 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 29091	. . . . . . . . . . . . . . 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 29095	. . . . . . . . . . . . . . 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 29014	. . . . . . . . . . . . . . . 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 29015	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ((((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇) ∨ 𝑉 = 𝑇))
167	157	neneqd 2961	. . . . . . . . . . . . . . . 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 29141	. . . . . . . . . . . . 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 2862	. . . . . . . . . . . 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 3011	. . . . . . . . . . . . . . . 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 28938	. . . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . 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 28938	. . . . . . . . . . . . . 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 28935	. . . . . . . . . . . . . . 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 28972	. . . . . . . . . . . . . 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 29321	. . . . . . . . . . . . 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 29063	. . . . . . . . . . . . . 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 29316	. . . . . . . . . . . 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 29322	. . . . . . . . . . 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 29063	. . . . . . . . . . 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 29316	. . . . . . . . . 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 29096	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
193	192	fveq2d 6887	. . . . . . . . . . . 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 29225	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄(((pInvG‘𝐺)‘𝑇)‘𝑈))
196	4, 28, 5, 149, 24, 158, 7, 17, 128, 195	oppcom 29213	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
197	4, 28, 5, 149, 24, 158, 7, 17, 124, 148	oppcom 29213	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊𝑄𝑈)
198	4, 5, 24, 149, 7, 158, 124, 128, 17, 197	lnopp2hpgb 29234	. . . . . . . . . . . . 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 29066	. . . . . . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . . . . . 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 3054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑅 ≠ 𝑍)
209	48, 37, 208	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ≠ 𝑍)
210	209	neneqd 2961	. . . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . . . . . . . . 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 29212	. . . . . . . . . . . . . . . . . . . . . . . 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 28938	. . . . . . . . . . . . . . . . . . . . . 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 29091	. . . . . . . . . . . . . . . . . . . . 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 29094	. . . . . . . . . . . . . . . . . . . . . 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 28938	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3011	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29085	. . . . . . . . . . . . . . . . . . . . . . . . 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 29094	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑉 ∈ (𝑉𝐿𝑇))
239		nelne2 3054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑉 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑈 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑈)
240	238, 152, 239	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
241	240	necomd 3011	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29091	. . . . . . . . . . . . . . . . . . . . . . . . . 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 29066	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29096	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29085	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29095	. . . . . . . . . . . . . . . . . . . . . . . . . 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 29097	. . . . . . . . . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . . . . . . . . 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 29084	. . . . . . . . . . . . . . . . . . . . . . 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 3053	. . . . . . . . . . . . . . . . . . . . . 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 29080	. . . . . . . . . . . . . . . . . . . . . 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 29095	. . . . . . . . . . . . . . . . . . . . . 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 29104	. . . . . . . . . . . . . . . . . . 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 28937	. . . . . . . . . . . . . . . . . 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 28944	. . . . . . . . . . . . . . . . 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 29209	. . . . . . . . . . . . . . . 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 29213	. . . . . . . . . . . . . . 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 29223	. . . . . . . . . . . . . 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 29213	. . . . . . . . . . . . 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 29234	. . . . . . . . . . . 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 29335	. . . . . . . . 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 29317	. . . . . . . 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 28919	. . . . . 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 3171	. . . . 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 3011	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
288	4, 5, 6, 19, 12, 49, 7, 60, 28, 287, 90	hlcgrex 29075	. . . . 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 3171	. . 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 29075	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 (𝑢(𝐾‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)))
293	291, 292	r19.29a 3171	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 2956 ∃wrex 3087 ∖ cdif 3896 class class class wbr 5103 {copab 5167 ran crn 5652 ‘cfv 6537 (class class class)co 7418 ⟨“cs3 14986 Basecbs 17380 distcds 17430 TarskiGcstrkg 28882 Itvcitv 28888 LineGclng 28889 cgrGccgrg 28966 hlGchlg 29056 pInvGcmir 29117 hpGchpg 29228 cgrAccgra 29307

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 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 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 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-oadd 8473 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-dju 9975 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-n0 12600 df-xnn0 12673 df-z 12687 df-uz 12959 df-fz 13633 df-fzo 13782 df-hash 14468 df-word 14652 df-concat 14709 df-s1 14736 df-s2 14992 df-s3 14993 df-trkgc 28903 df-trkgb 28904 df-trkgcb 28905 df-trkgld 28907 df-trkg 28908 df-cgrg 28967 df-leg 29039 df-hlg 29057 df-mir 29118 df-rag 29162 df-perpg 29164 df-hpg 29229 df-mid 29272 df-lmi 29273 df-cgra 29308

This theorem is used by: tgaaddcpbllem2 29343 tgaaddcpbllem3 29344

W3C validator