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Theorem tgaaddcpbllem1 29406

Description: Lemma for tgaaddcpbl 29409. (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 2772	. . . . . . . . 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 29158	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ (𝑌𝐿𝑆))
28		eqid 2763	. . . . . . . . . . . . . . 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 29155	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
31		tgaaddcpbl.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋𝑂𝑍)
32	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne1 29274	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
33		nelne2 3057	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑋)
34	27, 32, 33	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
35	34	necomd 3014	. . . . . . . . . . . 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 29275	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
38		nelne2 3057	. . . . . . . . . . . . 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 2763	. . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . 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 29069	. . . . . . . . . . . . . . . . 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 5123	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29386	. . . . . . . . . . . . . . . . . . . . . . . . 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 29381	. . . . . . . . . . . . . . . . . . . . . . 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 29386	. . . . . . . . . . . . . . . . . . . . . 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 29380	. . . . . . . . . . . . . . . . . . . . 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 29381	. . . . . . . . . . . . . . . . . . . 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 29132	. . . . . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑌 ∈ (𝑋𝐼𝑍))
88	83, 87	mtand 828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
89	88	neqned 2966	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
90	89	necomd 3014	. . . . . . . . . . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . . . . . . . . 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 29132	. . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . 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 29382	. . . . . . . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . . . . . . 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 3057	. . . . . . . . . . . . . . . . . 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 29002	. . . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . . . 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 28984	. . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . 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 29004	. . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . 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 29036	. . . . . . . . . . . 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 29048	. . . . . . . . . . 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 29385	. . . . . . . . . 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 29127	. . . . . . . . . 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 29380	. . . . . . . . 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 2763	. . . . . . . . . . . 12 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
127		eqid 2763	. . . . . . . . . . . 12 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
128	4, 28, 5, 24, 126, 7, 60, 127, 17	mircl 29190	. . . . . . . . . . 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 29160	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
138	135, 137	eleqtrd 2866	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑍𝐿𝑌))
139	39	necomd 3014	. . . . . . . . . . . . . . . 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 29148	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ (𝑌𝐿𝑆))
142	37, 141	mtand 828	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑆 ∈ (𝑌𝐿𝑍))
143	39	neneqd 2964	. . . . . . . . . . . . 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 29272	. . . . . . . . . . . . . 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 29191	. . . . . . . . . . . . 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 29155	. . . . . . . . . . . . . . 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 29159	. . . . . . . . . . . . . . 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 29078	. . . . . . . . . . . . . . . 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 29079	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ((((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇) ∨ 𝑉 = 𝑇))
167	157	neneqd 2964	. . . . . . . . . . . . . . . 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 29205	. . . . . . . . . . . . 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 2865	. . . . . . . . . . . 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 5123	. . . . . . . . . . 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 3014	. . . . . . . . . . . . . . . 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 29002	. . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . 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 29002	. . . . . . . . . . . . . 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 28999	. . . . . . . . . . . . . . 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 29036	. . . . . . . . . . . . . 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 29385	. . . . . . . . . . . . 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 29127	. . . . . . . . . . . . . 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 29380	. . . . . . . . . . . 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 29386	. . . . . . . . . . 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 29127	. . . . . . . . . . 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 29380	. . . . . . . . . 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 29160	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
193	192	fveq2d 6920	. . . . . . . . . . . 12 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
194	17, 152	eldifd 4053	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ (𝑃 ∖ (𝑉𝐿𝑇)))
195	4, 5, 126, 127, 149, 7, 158, 160, 194, 24	oppmir 29289	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄(((pInvG‘𝐺)‘𝑇)‘𝑈))
196	4, 28, 5, 149, 24, 158, 7, 17, 128, 195	oppcom 29277	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
197	4, 28, 5, 149, 24, 158, 7, 17, 124, 148	oppcom 29277	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊𝑄𝑈)
198	4, 5, 24, 149, 7, 158, 124, 128, 17, 197	lnopp2hpgb 29298	. . . . . . . . . . . . 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 5123	. . . . . . . . . . 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 29130	. . . . . . . . . . . . . . . . . 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 2867	. . . . . . . . . . . . . . . . 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 3057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑅 ≠ 𝑍)
209	48, 37, 208	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ≠ 𝑍)
210	209	neneqd 2964	. . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . 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 29276	. . . . . . . . . . . . . . . . . . . . . . . 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 29002	. . . . . . . . . . . . . . . . . . . . . 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 29155	. . . . . . . . . . . . . . . . . . . . 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 29158	. . . . . . . . . . . . . . . . . . . . . 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 29002	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3014	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29149	. . . . . . . . . . . . . . . . . . . . . . . . 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 29158	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑉 ∈ (𝑉𝐿𝑇))
239		nelne2 3057	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑉 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑈 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑈)
240	238, 152, 239	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
241	240	necomd 3014	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29155	. . . . . . . . . . . . . . . . . . . . . . . . . 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 29130	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29160	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29149	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29159	. . . . . . . . . . . . . . . . . . . . . . . . . 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 29161	. . . . . . . . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . . . . . . . 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 29148	. . . . . . . . . . . . . . . . . . . . . . 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 3056	. . . . . . . . . . . . . . . . . . . . . 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 29144	. . . . . . . . . . . . . . . . . . . . . 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 4100	. . . . . . . . . . . . . . . . . . . 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 29159	. . . . . . . . . . . . . . . . . . . . . 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 4100	. . . . . . . . . . . . . . . . . . . 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 29168	. . . . . . . . . . . . . . . . . . 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 29001	. . . . . . . . . . . . . . . . . 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 29008	. . . . . . . . . . . . . . . . 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 29273	. . . . . . . . . . . . . . . 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 29277	. . . . . . . . . . . . . . 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 29287	. . . . . . . . . . . . . 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 29277	. . . . . . . . . . . . 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 29298	. . . . . . . . . . . 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 5123	. . . . . . . . . 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 29399	. . . . . . . . 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 29381	. . . . . . . 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 5123	. . . . . . 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 28983	. . . . . 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 3174	. . . . 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 3014	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
288	4, 5, 6, 19, 12, 49, 7, 60, 28, 287, 90	hlcgrex 29139	. . . . 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 3174	. . 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 29139	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 (𝑢(𝐾‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)))
293	291, 292	r19.29a 3174	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 2147 ≠ wne 2959 ∃wrex 3090 ∖ cdif 4048 class class class wbr 5108 {copab 5173 ran crn 5659 ‘cfv 6566 (class class class)co 7451 ⟨“cs3 15035 Basecbs 17431 distcds 17481 TarskiGcstrkg 28946 Itvcitv 28952 LineGclng 28953 cgrGccgrg 29030 hlGchlg 29120 pInvGcmir 29181 hpGchpg 29292 cgrAccgra 29371

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-oadd 8518 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-dju 10019 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-nn 12376 df-2 12445 df-3 12446 df-n0 12647 df-xnn0 12720 df-z 12734 df-uz 13006 df-fz 13680 df-fzo 13829 df-hash 14516 df-word 14700 df-concat 14757 df-s1 14784 df-s2 15041 df-s3 15042 df-trkgc 28967 df-trkgb 28968 df-trkgcb 28969 df-trkgld 28971 df-trkg 28972 df-cgrg 29031 df-leg 29103 df-hlg 29121 df-mir 29182 df-rag 29226 df-perpg 29228 df-hpg 29293 df-mid 29336 df-lmi 29337 df-cgra 29372

This theorem is used by: tgaaddcpbllem2 29407 tgaaddcpbllem3 29408

W3C validator