tgaaddcpbllem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > tgaaddcpbllem1		Structured version Visualization version GIF version

Theorem tgaaddcpbllem1 29356

Description: Lemma for tgaaddcpbl 29359. (Contributed by Thierry Arnoux, 2-Aug-2026.)

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

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

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

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

Step	Hyp	Ref	Expression
1		tgaaddcpbl.c	. . . . . . . . . 10 ⊢ ∼ = (cgrA‘𝐺)
2	1	eqcomi 2769	. . . . . . . . 9 ⊢ (cgrA‘𝐺) = ∼
3	2	a1i 11	. . . . . . . 8 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (cgrA‘𝐺) = ∼ )
4		tgaaddcpbl.p	. . . . . . . . 9 ⊢ 𝑃 = (Base‘𝐺)
5		tgaaddcpbl.i	. . . . . . . . 9 ⊢ 𝐼 = (Itv‘𝐺)
6		tgaaddcpbllem1.1	. . . . . . . . 9 ⊢ 𝐾 = (hlG‘𝐺)
7		tgaaddcpbl.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8	7	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝐺 ∈ TarskiG)
9	8	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝐺 ∈ TarskiG)
10		tgaaddcpbl.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋 ∈ 𝑃)
12		tgaaddcpbl.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13	12	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ∈ 𝑃)
14		tgaaddcpbl.z	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
15	14	ad6antr 749	. . . . . . . . . 10 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑍 ∈ 𝑃)
16	15	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑍 ∈ 𝑃)
17		tgaaddcpbl.u	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
18	17	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑈 ∈ 𝑃)
19		tgaaddcpbl.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
20	19	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ∈ 𝑃)
21		simpllr 788	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑤 ∈ 𝑃)
22		simp-6r 800	. . . . . . . . . . 11 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑢 ∈ 𝑃)
23	22	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ∈ 𝑃)
24		tgaaddcpbl.l	. . . . . . . . . . . . . . 15 ⊢ 𝐿 = (LineG‘𝐺)
25		tgaaddcpbl.s	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
26		tgaaddcpbl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑌 ≠ 𝑆)
27	4, 5, 24, 7, 12, 25, 26	tglinerflx1 29108	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑌 ∈ (𝑌𝐿𝑆))
28		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (dist‘𝐺) = (dist‘𝐺)
29		tgaaddcpbl.o	. . . . . . . . . . . . . . 15 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}
30	4, 5, 24, 7, 12, 25, 26	tgelrnln 29105	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
31		tgaaddcpbl.4	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋𝑂𝑍)
32	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne1 29224	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
33		nelne2 3053	. . . . . . . . . . . . . 14 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑋)
34	27, 32, 33	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
35	34	necomd 3010	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
36	35	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋 ≠ 𝑌)
37	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne2 29225	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
38		nelne2 3053	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑍)
39	27, 37, 38	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
40	39	ad9antr 755	. . . . . . . . . . 11 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑍)
41		eqid 2760	. . . . . . . . . . . 12 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
42		simp-7r 802	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋))
43	42	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑋) = (𝑉(dist‘𝐺)𝑢))
44	4, 28, 5, 9, 13, 11, 20, 23, 43	tgcgrcomlr 28949	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑋(dist‘𝐺)𝑌) = (𝑢(dist‘𝐺)𝑉))
45	44	eqcomd 2766	. . . . . . . . . . . . 13 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢(dist‘𝐺)𝑉) = (𝑋(dist‘𝐺)𝑌))
46		simpllr 788	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → 𝑟 ∈ 𝑃)
47	46	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ 𝑃)
48		tgaaddcpbllem1.2	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑅 ∈ (𝑌𝐿𝑆))
49	4, 24, 5, 7, 30, 48	tglnpt 29019	. . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29336	. . . . . . . . . . . . . . . . . . . . . . . . 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 29331	. . . . . . . . . . . . . . . . . . . . . . 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 29336	. . . . . . . . . . . . . . . . . . . . . 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 29330	. . . . . . . . . . . . . . . . . . . . 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 29331	. . . . . . . . . . . . . . . . . . . 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 29082	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋(𝐾‘𝑌)𝑋)
81	80	ad9antr 755	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑋(𝐾‘𝑌)𝑋)
82	81	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑋(𝐾‘𝑌)𝑋)
83		tgaaddcpbllem3.1	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ¬ 𝑌 ∈ (𝑋𝐼𝑍))
84		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑅 = 𝑌)
85		tgaaddcpbllem1.3	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑅 ∈ (𝑋𝐼𝑍))
86	85	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑅 ∈ (𝑋𝐼𝑍))
87	84, 86	eqeltrrd 2861	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑅 = 𝑌) → 𝑌 ∈ (𝑋𝐼𝑍))
88	83, 87	mtand 828	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ¬ 𝑅 = 𝑌)
89	88	neqned 2962	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝑅 ≠ 𝑌)
90	89	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑌 ≠ 𝑅)
91	90	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑌 ≠ 𝑅)
92	91	necomd 3010	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅 ≠ 𝑌)
93	4, 5, 6, 55, 54, 58, 53, 92	hlid 29082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅(𝐾‘𝑌)𝑅)
94	43	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑌(dist‘𝐺)𝑋) = (𝑉(dist‘𝐺)𝑢))
95		simp-4r 796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅))
96	95	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑅) = (𝑉(dist‘𝐺)𝑟))
97	96	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑌(dist‘𝐺)𝑅) = (𝑉(dist‘𝐺)𝑟))
98	4, 5, 6, 53, 54, 58, 55, 56, 59, 57, 79, 54, 28, 55, 82, 93, 94, 97	cgracgr 29332	. . . . . . . . . . . . . . . . . . 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 28949	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → (𝑅(dist‘𝐺)𝑋) = (𝑟(dist‘𝐺)𝑢))
100	52, 99	mpdan 700	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑅(dist‘𝐺)𝑋) = (𝑟(dist‘𝐺)𝑢))
101	48	ad10antr 757	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑟(𝐾‘𝑉)𝑇) → 𝑅 ∈ (𝑌𝐿𝑆))
102	52, 101	mpdan 700	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ∈ (𝑌𝐿𝑆))
103	32	ad9antr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
104		nelne2 3053	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑅 ≠ 𝑋)
105	102, 103, 104	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ≠ 𝑋)
106	4, 28, 5, 9, 51, 11, 47, 23, 100, 105	tgcgrneq 28952	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ≠ 𝑢)
107	106	necomd 3010	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ≠ 𝑟)
108		simplr 781	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑢𝐼𝑤))
109	85	ad9antr 755	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ∈ (𝑋𝐼𝑍))
110	100	eqcomd 2766	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑟(dist‘𝐺)𝑢) = (𝑅(dist‘𝐺)𝑋))
111	4, 28, 5, 9, 47, 23, 51, 11, 110	tgcgrcomlr 28949	. . . . . . . . . . . . . . 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 28949	. . . . . . . . . . . . . . 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 28934	. . . . . . . . . . . . . 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 28949	. . . . . . . . . . . . 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 28954	. . . . . . . . . . . . . 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 28949	. . . . . . . . . . . . 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 28986	. . . . . . . . . . . 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 28998	. . . . . . . . . . 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 29335	. . . . . . . . . 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 29077	. . . . . . . . . 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 29330	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑤”⟩)
124		tgaaddcpbl.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
125	124	ad9antr 755	. . . . . . . . 9 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑊 ∈ 𝑃)
126		eqid 2760	. . . . . . . . . . . 12 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
127		eqid 2760	. . . . . . . . . . . 12 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
128	4, 28, 5, 24, 126, 7, 60, 127, 17	mircl 29140	. . . . . . . . . . 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 29110	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
138	135, 137	eleqtrd 2862	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑍𝐿𝑌))
139	39	necomd 3010	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
140	139	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ≠ 𝑌)
141	4, 5, 24, 130, 131, 132, 133, 134, 138, 140	lnrot1 29098	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ (𝑌𝐿𝑆))
142	37, 141	mtand 828	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑆 ∈ (𝑌𝐿𝑍))
143	39	neneqd 2960	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑌 = 𝑍)
144	142, 143	jca 521	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ 𝑆 ∈ (𝑌𝐿𝑍) ∧ ¬ 𝑌 = 𝑍))
145		ioran 999	. . . . . . . . . . . 12 ⊢ (¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ (¬ 𝑆 ∈ (𝑌𝐿𝑍) ∧ ¬ 𝑌 = 𝑍))
146	144, 145	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
147	146	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
148		tgaaddcpbl.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄𝑊)
149		tgaaddcpbl.q	. . . . . . . . . . . . . . 15 ⊢ 𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}
150	4, 28, 5, 149, 17, 124	islnopp 29222	. . . . . . . . . . . . . 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 29141	. . . . . . . . . . . . 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 29105	. . . . . . . . . . . . . . 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 29109	. . . . . . . . . . . . . . 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 29028	. . . . . . . . . . . . . . . 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 29029	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ((((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇) ∨ 𝑉 = 𝑇))
167	157	neneqd 2960	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ¬ 𝑉 = 𝑇)
168	167	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ¬ 𝑉 = 𝑇)
169	166, 168	olcnd 891	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇))
170	4, 28, 5, 24, 126, 153, 127, 159, 161, 169	mirln 29155	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∈ (𝑉𝐿𝑇))
171	156, 170	eqeltrrd 2861	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑈 ∈ (𝑉𝐿𝑇))
172	152, 171	mtand 828	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
173	172	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
174		tgaaddcpbl.7	. . . . . . . . . . . 12 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
175	66, 174	breqdi 5117	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
176	175	ad9antr 755	. . . . . . . . . 10 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
177	52, 92	mpdan 700	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑅 ≠ 𝑌)
178	177	necomd 3010	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑅)
179	4, 28, 5, 9, 13, 51, 20, 47, 96, 178	tgcgrneq 28952	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑟)
180	179	necomd 3010	. . . . . . . . . . . . . 14 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ≠ 𝑉)
181	115	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑍) = (𝑉(dist‘𝐺)𝑤))
182	4, 28, 5, 9, 13, 16, 20, 21, 181, 40	tgcgrneq 28952	. . . . . . . . . . . . . 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 28949	. . . . . . . . . . . . . . 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 28986	. . . . . . . . . . . . . 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 29335	. . . . . . . . . . . . 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 29077	. . . . . . . . . . . . . 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 29330	. . . . . . . . . . . 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 29336	. . . . . . . . . . 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 29077	. . . . . . . . . . 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 29330	. . . . . . . . . 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 29110	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
193	192	fveq2d 6892	. . . . . . . . . . . 12 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
194	17, 152	eldifd 4049	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ (𝑃 ∖ (𝑉𝐿𝑇)))
195	4, 5, 126, 127, 149, 7, 158, 160, 194, 24	oppmir 29239	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄(((pInvG‘𝐺)‘𝑇)‘𝑈))
196	4, 28, 5, 149, 24, 158, 7, 17, 128, 195	oppcom 29227	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
197	4, 28, 5, 149, 24, 158, 7, 17, 124, 148	oppcom 29227	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑊𝑄𝑈)
198	4, 5, 24, 149, 7, 158, 124, 128, 17, 197	lnopp2hpgb 29248	. . . . . . . . . . . . 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 5117	. . . . . . . . . . 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 29080	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑇𝐿𝑉))
206	192	ad9antr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
207	205, 206	eleqtrrd 2863	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑟 ∈ (𝑉𝐿𝑇))
208		nelne2 3053	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑅 ≠ 𝑍)
209	48, 37, 208	syl2anc 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑅 ≠ 𝑍)
210	209	neneqd 2960	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ¬ 𝑅 = 𝑍)
211	210	ad9antr 755	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑅 = 𝑍)
212	9	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝐺 ∈ TarskiG)
213	47	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑟 ∈ 𝑃)
214	21	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ 𝑃)
215	51	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑅 ∈ 𝑃)
216	16	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑍 ∈ 𝑃)
217	112	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍))
218	116	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑋(dist‘𝐺)𝑍) = (𝑢(dist‘𝐺)𝑤))
219	4, 28, 5, 29, 24, 30, 7, 10, 14, 31	oppne3 29226	. . . . . . . . . . . . . . . . . . . . . . . 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 28952	. . . . . . . . . . . . . . . . . . . . . 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 29105	. . . . . . . . . . . . . . . . . . . . 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 29108	. . . . . . . . . . . . . . . . . . . . . 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 28952	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑢)
234	233	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ≠ 𝑉)
235	234	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ≠ 𝑉)
236		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ (𝑉𝐿𝑇))
237	4, 5, 24, 226, 231, 227, 228, 235, 236, 230	lnrot2 29099	. . . . . . . . . . . . . . . . . . . . . . . . 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 29108	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑉 ∈ (𝑉𝐿𝑇))
239		nelne2 3053	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑉 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑈 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑈)
240	238, 152, 239	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
241	240	necomd 3010	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
242	241	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ≠ 𝑉)
243	4, 5, 24, 226, 231, 227, 235	tgelrnln 29105	. . . . . . . . . . . . . . . . . . . . . . . . . 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 29080	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 29110	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑈𝐿𝑉) = (𝑉𝐿𝑈))
247	244, 246	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑢 ∈ (𝑉𝐿𝑈))
248	240	ad9antr 755	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑈)
249	4, 5, 24, 9, 23, 20, 18, 234, 247, 248	lnrot2 29099	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 29109	. . . . . . . . . . . . . . . . . . . . . . . . . 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 29111	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑉) = (𝑈𝐿𝑉))
253	237, 252	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ (𝑈𝐿𝑉))
254	4, 5, 24, 226, 227, 228, 229, 230, 253, 242	lnrot1 29098	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ (𝑉𝐿𝑇))
255	152	ad10antr 757	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → ¬ 𝑈 ∈ (𝑉𝐿𝑇))
256	254, 255	pm2.65da 829	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → ¬ 𝑢 ∈ (𝑉𝐿𝑇))
257		nelne1 3052	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑢 ∈ (𝑢𝐿𝑤) ∧ ¬ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
258	225, 256, 257	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
259	258	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑟 ∈ (𝑢𝐼𝑤)) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
260	4, 5, 24, 9, 23, 21, 47, 221, 108	btwnlng1 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)) → 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 4096	. . . . . . . . . . . . . . . . . . . 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 29109	. . . . . . . . . . . . . . . . . . . . . 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 4096	. . . . . . . . . . . . . . . . . . . 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 29118	. . . . . . . . . . . . . . . . . . 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 28951	. . . . . . . . . . . . . . . . . 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 28958	. . . . . . . . . . . . . . . . 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 29223	. . . . . . . . . . . . . . . 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 29227	. . . . . . . . . . . . . . 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 29237	. . . . . . . . . . . . . 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 29227	. . . . . . . . . . . . 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 29248	. . . . . . . . . . . 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 5117	. . . . . . . . . 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 29349	. . . . . . . . 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 29331	. . . . . . . 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 5117	. . . . . . 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 28933	. . . . . 6 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → ∃𝑤 ∈ 𝑃 (𝑟 ∈ (𝑢𝐼𝑤) ∧ (𝑟(dist‘𝐺)𝑤) = (𝑅(dist‘𝐺)𝑍)))
285	283, 284	r19.29a 3170	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ 𝑟(𝐾‘𝑉)𝑇) ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
286	285	anasss 472	. . . 4 ⊢ ((((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑟 ∈ 𝑃) ∧ (𝑟(𝐾‘𝑉)𝑇 ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
287	157	necomd 3010	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 𝑉)
288	4, 5, 6, 19, 12, 49, 7, 60, 28, 287, 90	hlcgrex 29089	. . . . 5 ⊢ (𝜑 → ∃𝑟 ∈ 𝑃 (𝑟(𝐾‘𝑉)𝑇 ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)))
289	288	ad3antrrr 743	. . . 4 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ∃𝑟 ∈ 𝑃 (𝑟(𝐾‘𝑉)𝑇 ∧ (𝑉(dist‘𝐺)𝑟) = (𝑌(dist‘𝐺)𝑅)))
290	286, 289	r19.29a 3170	. . 3 ⊢ ((((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ 𝑢(𝐾‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
291	290	anasss 472	. 2 ⊢ (((𝜑 ∧ 𝑢 ∈ 𝑃) ∧ (𝑢(𝐾‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
292	4, 5, 6, 19, 12, 10, 7, 17, 28, 241, 34	hlcgrex 29089	. 2 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 (𝑢(𝐾‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)))
293	291, 292	r19.29a 3170	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 ∖ cdif 4044 class class class wbr 5102 {copab 5166 ran crn 5651 ‘cfv 6541 (class class class)co 7422 ⟨“cs3 14999 Basecbs 17394 distcds 17444 TarskiGcstrkg 28896 Itvcitv 28902 LineGclng 28903 cgrGccgrg 28980 hlGchlg 29070 pInvGcmir 29131 hpGchpg 29242 cgrAccgra 29321

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-1cn 11263 ax-icn 11264 ax-addcl 11265 ax-addrcl 11266 ax-mulcl 11267 ax-mulrcl 11268 ax-mulcom 11269 ax-addass 11270 ax-mulass 11271 ax-distr 11272 ax-i2m1 11273 ax-1ne0 11274 ax-1rid 11275 ax-rnegex 11276 ax-rrecex 11277 ax-cnre 11278 ax-pre-lttri 11279 ax-pre-lttrn 11280 ax-pre-ltadd 11281 ax-pre-mulgt0 11282

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7379 df-ov 7425 df-oprab 7426 df-mpo 7427 df-om 7880 df-1st 8003 df-2nd 8004 df-frecs 8304 df-wrecs 8335 df-recs 8384 df-rdg 8423 df-1o 8481 df-oadd 8485 df-er 8722 df-map 8854 df-pm 8855 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-dju 9987 df-card 10025 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-le 11354 df-sub 11548 df-neg 11549 df-nn 12341 df-2 12410 df-3 12411 df-n0 12612 df-xnn0 12685 df-z 12699 df-uz 12971 df-fz 13645 df-fzo 13794 df-hash 14480 df-word 14664 df-concat 14721 df-s1 14748 df-s2 15005 df-s3 15006 df-trkgc 28917 df-trkgb 28918 df-trkgcb 28919 df-trkgld 28921 df-trkg 28922 df-cgrg 28981 df-leg 29053 df-hlg 29071 df-mir 29132 df-rag 29176 df-perpg 29178 df-hpg 29243 df-mid 29286 df-lmi 29287 df-cgra 29322

This theorem is used by: tgaaddcpbllem2 29357 tgaaddcpbllem3 29358

W3C validator