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Theorem tgaaddcpbl 29231

Description: The angular addition is compatible with angle congruence: by adding congruent angles together, we obtain congruent angles. Theorem 11.22 of [Schwabhauser] p. 99. The angles ⟨“𝑋𝑌𝑆”⟩ and ⟨“𝑆𝑌𝑍”⟩ are added to result in ⟨“𝑋𝑌𝑍”⟩, and ⟨“𝑈𝑉𝑇”⟩ and ⟨“𝑇𝑉𝑊”⟩ are added to result in ⟨“𝑈𝑉𝑊”⟩. (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”⟩)

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

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

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

Step	Hyp	Ref	Expression
1		tgaaddcpbl.c	. . . . . . . . 9 ⊢ ∼ = (cgrA‘𝐺)
2	1	a1i 11	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ∼ = (cgrA‘𝐺))
3	2	eqcomd 2766	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → (cgrA‘𝐺) = ∼ )
4		tgaaddcpbl.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
5		tgaaddcpbl.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
6		eqid 2760	. . . . . . . 8 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
7		tgaaddcpbl.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
8	7	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝐺 ∈ TarskiG)
9	8	ad3antrrr 743	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝐺 ∈ TarskiG)
10		tgaaddcpbl.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
11	10	ad7antr 751	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑋 ∈ 𝑃)
12		tgaaddcpbl.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
13	12	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑌 ∈ 𝑃)
14	13	ad3antrrr 743	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑌 ∈ 𝑃)
15		tgaaddcpbl.z	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
16	15	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑍 ∈ 𝑃)
17	16	ad3antrrr 743	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑍 ∈ 𝑃)
18		tgaaddcpbl.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
19	18	ad7antr 751	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑈 ∈ 𝑃)
20		tgaaddcpbl.v	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
21	20	ad4antr 745	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑉 ∈ 𝑃)
22	21	ad3antrrr 743	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑉 ∈ 𝑃)
23		simpllr 788	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑤 ∈ 𝑃)
24		eqid 2760	. . . . . . . . 9 ⊢ (dist‘𝐺) = (dist‘𝐺)
25		simp-7r 802	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑌 ∈ (𝑋𝐼𝑍))
26		simpllr 788	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑢 ∈ 𝑃)
27	26	ad3antrrr 743	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑢 ∈ 𝑃)
28		simp-5r 798	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑢((hlG‘𝐺)‘𝑉)𝑈)
29		simplr 781	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑉 ∈ (𝑢𝐼𝑤))
30	4, 5, 6, 27, 19, 23, 9, 22, 28, 29	btwnhl 28959	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑉 ∈ (𝑈𝐼𝑤))
31		tgaaddcpbl.l	. . . . . . . . . . . . 13 ⊢ 𝐿 = (LineG‘𝐺)
32		tgaaddcpbl.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
33		tgaaddcpbl.2	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ≠ 𝑆)
34	4, 5, 31, 7, 12, 32, 33	tglinerflx1 28980	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑌𝐿𝑆))
35		tgaaddcpbl.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝑌𝐿𝑆)) ∧ 𝑏 ∈ (𝑃 ∖ (𝑌𝐿𝑆))) ∧ ∃𝑠 ∈ (𝑌𝐿𝑆)𝑠 ∈ (𝑎𝐼𝑏))}
36	4, 5, 31, 7, 12, 32, 33	tgelrnln 28977	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
37		tgaaddcpbl.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋𝑂𝑍)
38	4, 24, 5, 35, 31, 36, 7, 10, 15, 37	oppne1 29096	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
39		nelne2 3053	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑋)
40	34, 38, 39	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
41	40	necomd 3010	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
42	41	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑋 ≠ 𝑌)
43	4, 24, 5, 35, 31, 36, 7, 10, 15, 37	oppne2 29097	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
44		nelne2 3053	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑍)
45	34, 43, 44	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
46	45	necomd 3010	. . . . . . . . . 10 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
47	46	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑍 ≠ 𝑌)
48		tgaaddcpbl.t	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ 𝑃)
49		tgaaddcpbl.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑉 ≠ 𝑇)
50	4, 5, 31, 7, 20, 48, 49	tglinerflx1 28980	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑉 ∈ (𝑉𝐿𝑇))
51		tgaaddcpbl.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈𝑄𝑊)
52		tgaaddcpbl.q	. . . . . . . . . . . . . . 15 ⊢ 𝑄 = {⟨𝑐, 𝑑⟩ ∣ ((𝑐 ∈ (𝑃 ∖ (𝑉𝐿𝑇)) ∧ 𝑑 ∈ (𝑃 ∖ (𝑉𝐿𝑇))) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑐𝐼𝑑))}
53		tgaaddcpbl.w	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
54	4, 24, 5, 52, 18, 53	islnopp 29094	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑈𝑄𝑊 ↔ ((¬ 𝑈 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑊 ∈ (𝑉𝐿𝑇)) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑈𝐼𝑊))))
55	51, 54	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((¬ 𝑈 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑊 ∈ (𝑉𝐿𝑇)) ∧ ∃𝑡 ∈ (𝑉𝐿𝑇)𝑡 ∈ (𝑈𝐼𝑊)))
56	55	simplld 780	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑈 ∈ (𝑉𝐿𝑇))
57		nelne2 3053	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑈 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑈)
58	50, 56, 57	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
59	58	necomd 3010	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
60	59	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑈 ≠ 𝑉)
61		simpr 490	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍))
62	61	eqcomd 2766	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → (𝑌(dist‘𝐺)𝑍) = (𝑉(dist‘𝐺)𝑤))
63	45	ad7antr 751	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑌 ≠ 𝑍)
64	4, 24, 5, 9, 14, 17, 22, 23, 62, 63	tgcgrneq 28824	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑉 ≠ 𝑤)
65	64	necomd 3010	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑤 ≠ 𝑉)
66	4, 5, 24, 9, 11, 14, 17, 19, 22, 23, 25, 30, 42, 47, 60, 65	flatcgra 29211	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑤”⟩)
67	53	ad7antr 751	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑊 ∈ 𝑃)
68	32	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑆 ∈ 𝑃)
69	48	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑇 ∈ 𝑃)
70		eqid 2760	. . . . . . . . . . 11 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
71		eqid 2760	. . . . . . . . . . 11 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
72	4, 24, 5, 31, 70, 7, 48, 71, 18	mircl 29012	. . . . . . . . . 10 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ 𝑃)
73	72	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ 𝑃)
74	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝐺 ∈ TarskiG)
75	12	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑌 ∈ 𝑃)
76	32	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ 𝑃)
77	15	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ 𝑃)
78	33	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑌 ≠ 𝑆)
79		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑌𝐿𝑍))
80	45	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑌 ≠ 𝑍)
81	4, 5, 31, 74, 75, 77, 80	tglinecom 28982	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
82	79, 81	eleqtrd 2862	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑆 ∈ (𝑍𝐿𝑌))
83	46	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ≠ 𝑌)
84	4, 5, 31, 74, 75, 76, 77, 78, 82, 83	lnrot1 28970	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ (𝑌𝐿𝑆))
85	43, 84	mtand 828	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑆 ∈ (𝑌𝐿𝑍))
86	45	neneqd 2960	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑌 = 𝑍)
87		ioran 999	. . . . . . . . . . 11 ⊢ (¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍) ↔ (¬ 𝑆 ∈ (𝑌𝐿𝑍) ∧ ¬ 𝑌 = 𝑍))
88	85, 86, 87	sylanbrc 595	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
89	88	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ¬ (𝑆 ∈ (𝑌𝐿𝑍) ∨ 𝑌 = 𝑍))
90	7	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝐺 ∈ TarskiG)
91	48	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑇 ∈ 𝑃)
92	18	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑈 ∈ 𝑃)
93	4, 24, 5, 31, 70, 90, 91, 71, 92	mirmir 29013	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘(((pInvG‘𝐺)‘𝑇)‘𝑈)) = 𝑈)
94	4, 5, 31, 7, 20, 48, 49	tgelrnln 28977	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑉𝐿𝑇) ∈ ran 𝐿)
95	94	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (𝑉𝐿𝑇) ∈ ran 𝐿)
96	4, 5, 31, 7, 20, 48, 49	tglinerflx2 28981	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇 ∈ (𝑉𝐿𝑇))
97	96	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑇 ∈ (𝑉𝐿𝑇))
98	72	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ 𝑃)
99	20	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑉 ∈ 𝑃)
100		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
101	4, 31, 5, 90, 99, 98, 91, 100	colcom 28900	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (𝑇 ∈ ((((pInvG‘𝐺)‘𝑇)‘𝑈)𝐿𝑉) ∨ (((pInvG‘𝐺)‘𝑇)‘𝑈) = 𝑉))
102	4, 31, 5, 90, 98, 99, 91, 101	colrot1 28901	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ((((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇) ∨ 𝑉 = 𝑇))
103	49	neneqd 2960	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑉 = 𝑇)
104	103	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ¬ 𝑉 = 𝑇)
105	102, 104	olcnd 891	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇))
106	4, 24, 5, 31, 70, 90, 71, 95, 97, 105	mirln 29027	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → (((pInvG‘𝐺)‘𝑇)‘(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∈ (𝑉𝐿𝑇))
107	93, 106	eqeltrrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → 𝑈 ∈ (𝑉𝐿𝑇))
108	56, 107	mtand 828	. . . . . . . . . 10 ⊢ (𝜑 → ¬ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
109	108	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ¬ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈)))
110	1	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∼ = (cgrA‘𝐺))
111		tgaaddcpbl.7	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
112	110, 111	breqdi 5118	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
113	112	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑊”⟩)
114		tgaaddcpbl.6	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
115	110, 114	breqdi 5118	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
116	4, 5, 7, 6, 10, 12, 32, 18, 20, 48, 115	cgracom 29208	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝑈𝑉𝑇”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑆”⟩)
117	116	ad7antr 751	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑈𝑉𝑇”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑆”⟩)
118	4, 5, 24, 9, 19, 22, 69, 11, 14, 68, 23, 17, 117, 30, 25, 64, 63	sacgr 29218	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑤𝑉𝑇”⟩(cgrA‘𝐺)⟨“𝑍𝑌𝑆”⟩)
119	4, 5, 24, 9, 23, 22, 69, 17, 14, 68, 118	cgraswaplr 29212	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑇𝑉𝑤”⟩(cgrA‘𝐺)⟨“𝑆𝑌𝑍”⟩)
120	4, 5, 9, 6, 69, 22, 23, 68, 14, 17, 119	cgracom 29208	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑆𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑇𝑉𝑤”⟩)
121	4, 5, 31, 7, 20, 48, 49	tglinecom 28982	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
122	121	fveq2d 6882	. . . . . . . . . . 11 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
123	18, 56	eldifd 3910	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (𝑃 ∖ (𝑉𝐿𝑇)))
124	4, 5, 70, 71, 52, 7, 94, 96, 123, 31	oppmir 29111	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈𝑄(((pInvG‘𝐺)‘𝑇)‘𝑈))
125	4, 24, 5, 52, 31, 94, 7, 18, 72, 124	oppcom 29099	. . . . . . . . . . . 12 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
126	4, 24, 5, 52, 31, 94, 7, 18, 53, 51	oppcom 29099	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊𝑄𝑈)
127	4, 5, 31, 52, 7, 94, 53, 72, 18, 126	lnopp2hpgb 29120	. . . . . . . . . . . 12 ⊢ (𝜑 → ((((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈 ↔ 𝑊((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈)))
128	125, 127	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈))
129	122, 128	breqdi 5118	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊((hpG‘𝐺)‘(𝑇𝐿𝑉))(((pInvG‘𝐺)‘𝑇)‘𝑈))
130	129	ad7antr 751	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑊((hpG‘𝐺)‘(𝑇𝐿𝑉))(((pInvG‘𝐺)‘𝑇)‘𝑈))
131	122	ad7antr 751	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
132	125	ad7antr 751	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
133	94	ad7antr 751	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → (𝑉𝐿𝑇) ∈ ran 𝐿)
134	50	ad7antr 751	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑉 ∈ (𝑉𝐿𝑇))
135	8	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝐺 ∈ TarskiG)
136	21	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ 𝑃)
137	48	ad5antr 747	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ 𝑃)
138	18	ad5antr 747	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ 𝑃)
139	49	ad5antr 747	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑇)
140	26	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ 𝑃)
141	10	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑋 ∈ 𝑃)
142		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋))
143	142	eqcomd 2766	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → (𝑌(dist‘𝐺)𝑋) = (𝑉(dist‘𝐺)𝑢))
144	40	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑌 ≠ 𝑋)
145	4, 24, 5, 8, 13, 141, 21, 26, 143, 144	tgcgrneq 28824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑉 ≠ 𝑢)
146	145	necomd 3010	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑢 ≠ 𝑉)
147	146	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ≠ 𝑉)
148		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ (𝑉𝐿𝑇))
149	4, 5, 31, 135, 140, 136, 137, 147, 148, 139	lnrot2 28971	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ (𝑢𝐿𝑉))
150	59	ad5antr 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ≠ 𝑉)
151	4, 5, 31, 135, 140, 136, 147	tgelrnln 28977	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑉) ∈ ran 𝐿)
152	18	ad4antr 745	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑈 ∈ 𝑃)
153		simplr 781	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑢((hlG‘𝐺)‘𝑉)𝑈)
154	4, 5, 6, 26, 152, 21, 8, 153	hlcomd 28949	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑈((hlG‘𝐺)‘𝑉)𝑢)
155	4, 5, 6, 152, 26, 21, 8, 31, 154	hlln 28952	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑈 ∈ (𝑢𝐿𝑉))
156	155	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ (𝑢𝐿𝑉))
157	4, 5, 31, 135, 140, 136, 147	tglinerflx2 28981	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ (𝑢𝐿𝑉))
158	4, 5, 31, 135, 138, 136, 150, 150, 151, 156, 157	tglinethru 28983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑉) = (𝑈𝐿𝑉))
159	149, 158	eleqtrd 2862	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑇 ∈ (𝑈𝐿𝑉))
160	4, 5, 31, 135, 136, 137, 138, 139, 159, 150	lnrot1 28970	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → 𝑈 ∈ (𝑉𝐿𝑇))
161	56	ad5antr 747	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑢 ∈ (𝑉𝐿𝑇)) → ¬ 𝑈 ∈ (𝑉𝐿𝑇))
162	160, 161	pm2.65da 829	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ¬ 𝑢 ∈ (𝑉𝐿𝑇))
163	162	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ¬ 𝑢 ∈ (𝑉𝐿𝑇))
164	64	neneqd 2960	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ¬ 𝑉 = 𝑤)
165	9	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝐺 ∈ TarskiG)
166	27	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ 𝑃)
167	23	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ 𝑃)
168	9	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝐺 ∈ TarskiG)
169	23	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑤 ∈ 𝑃)
170	22	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑉 ∈ 𝑃)
171		simpllr 788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑉 ∈ (𝑢𝐼𝑤))
172		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑢 = 𝑤)
173	172	oveq1d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → (𝑢𝐼𝑤) = (𝑤𝐼𝑤))
174	171, 173	eleqtrd 2862	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑉 ∈ (𝑤𝐼𝑤))
175	4, 24, 5, 168, 169, 170, 174	axtgbtwnid 28807	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑤 = 𝑉)
176	175	eqcomd 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑢 = 𝑤) → 𝑉 = 𝑤)
177	64, 176	mteqand 3046	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑢 ≠ 𝑤)
178	177	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑢 ≠ 𝑤)
179	4, 5, 31, 165, 166, 167, 178	tgelrnln 28977	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ∈ ran 𝐿)
180	133	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑉𝐿𝑇) ∈ ran 𝐿)
181	4, 5, 31, 165, 166, 167, 178	tglinerflx1 28980	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑢 ∈ (𝑢𝐿𝑤))
182	163	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → ¬ 𝑢 ∈ (𝑉𝐿𝑇))
183		nelne1 3052	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑢 ∈ (𝑢𝐿𝑤) ∧ ¬ 𝑢 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
184	181, 182, 183	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → (𝑢𝐿𝑤) ≠ (𝑉𝐿𝑇))
185	22	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ 𝑃)
186		simpllr 788	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ (𝑢𝐼𝑤))
187	4, 5, 31, 165, 166, 167, 185, 178, 186	btwnlng1 28966	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ (𝑢𝐿𝑤))
188	134	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ (𝑉𝐿𝑇))
189	187, 188	elind 4146	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑉 ∈ ((𝑢𝐿𝑤) ∩ (𝑉𝐿𝑇)))
190	4, 5, 31, 165, 166, 167, 178	tglinerflx2 28981	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ (𝑢𝐿𝑤))
191		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ (𝑉𝐿𝑇))
192	190, 191	elind 4146	. . . . . . . . . . . . . . . . 17 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑤 ∈ ((𝑢𝐿𝑤) ∩ (𝑉𝐿𝑇)))
193	4, 5, 31, 165, 179, 180, 184, 189, 192	tglineineq 28990	. . . . . . . . . . . . . . . 16 ⊢ (((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) ∧ 𝑤 ∈ (𝑉𝐿𝑇)) → 𝑉 = 𝑤)
194	164, 193	mtand 828	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ¬ 𝑤 ∈ (𝑉𝐿𝑇))
195	4, 24, 5, 52, 27, 23, 134, 163, 194, 29	islnoppd 29095	. . . . . . . . . . . . . 14 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑢𝑄𝑤)
196	4, 24, 5, 52, 31, 133, 9, 6, 27, 19, 23, 195, 134, 28	opphl 29109	. . . . . . . . . . . . 13 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑈𝑄𝑤)
197	4, 24, 5, 52, 31, 133, 9, 19, 23, 196	oppcom 29099	. . . . . . . . . . . 12 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑤𝑄𝑈)
198	4, 5, 31, 52, 9, 133, 23, 73, 19, 197	lnopp2hpgb 29120	. . . . . . . . . . 11 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ((((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈 ↔ 𝑤((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈)))
199	132, 198	mpbid 235	. . . . . . . . . 10 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑤((hpG‘𝐺)‘(𝑉𝐿𝑇))(((pInvG‘𝐺)‘𝑇)‘𝑈))
200	131, 199	breqdi 5118	. . . . . . . . 9 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑤((hpG‘𝐺)‘(𝑇𝐿𝑉))(((pInvG‘𝐺)‘𝑇)‘𝑈))
201	4, 5, 24, 9, 68, 14, 17, 69, 22, 73, 31, 89, 109, 67, 23, 6, 113, 120, 130, 200	acopyeu 29221	. . . . . . . 8 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → 𝑊((hlG‘𝐺)‘𝑉)𝑤)
202	4, 5, 6, 9, 11, 14, 17, 19, 22, 23, 66, 67, 201	cgrahl2 29203	. . . . . . 7 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑊”⟩)
203	3, 202	breqdi 5118	. . . . . 6 ⊢ ((((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ 𝑉 ∈ (𝑢𝐼𝑤)) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
204	203	anasss 472	. . . . 5 ⊢ (((((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) ∧ 𝑤 ∈ 𝑃) ∧ (𝑉 ∈ (𝑢𝐼𝑤) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
205	4, 24, 5, 8, 26, 21, 13, 16	axtgsegcon 28805	. . . . 5 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ∃𝑤 ∈ 𝑃 (𝑉 ∈ (𝑢𝐼𝑤) ∧ (𝑉(dist‘𝐺)𝑤) = (𝑌(dist‘𝐺)𝑍)))
206	204, 205	r19.29a 3170	. . . 4 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
207	206	anasss 472	. . 3 ⊢ ((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ (𝑢((hlG‘𝐺)‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋))) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
208	4, 5, 6, 20, 12, 10, 7, 18, 24, 59, 40	hlcgrex 28961	. . . 4 ⊢ (𝜑 → ∃𝑢 ∈ 𝑃 (𝑢((hlG‘𝐺)‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)))
209	208	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ∃𝑢 ∈ 𝑃 (𝑢((hlG‘𝐺)‘𝑉)𝑈 ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)))
210	207, 209	r19.29a 3170	. 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
211	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝐺 ∈ TarskiG)
212	32	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑆 ∈ 𝑃)
213	48	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑇 ∈ 𝑃)
214	18	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑈 ∈ 𝑃)
215	20	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑉 ∈ 𝑃)
216	53	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑊 ∈ 𝑃)
217	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋 ∈ 𝑃)
218	12	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ∈ 𝑃)
219	15	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑍 ∈ 𝑃)
220	33	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑌 ≠ 𝑆)
221	49	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑉 ≠ 𝑇)
222	37	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑋𝑂𝑍)
223	51	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → 𝑈𝑄𝑊)
224	114	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑋𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑇”⟩)
225	111	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑆𝑌𝑍”⟩ ∼ ⟨“𝑇𝑉𝑊”⟩)
226		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → ¬ 𝑌 ∈ (𝑋𝐼𝑍))
227	4, 5, 31, 1, 35, 52, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226	tgaaddcpbllem3 29230	. 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
228	210, 227	pm2.61dan 825	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

Colors of variables: wff setvar class

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

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

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-dju 9906 df-card 9944 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12888 df-fz 13562 df-fzo 13710 df-hash 14395 df-word 14579 df-concat 14636 df-s1 14663 df-s2 14919 df-s3 14920 df-trkgc 28789 df-trkgb 28790 df-trkgcb 28791 df-trkgld 28793 df-trkg 28794 df-cgrg 28853 df-leg 28925 df-hlg 28943 df-mir 29004 df-rag 29048 df-perpg 29050 df-hpg 29115 df-mid 29158 df-lmi 29159 df-cgra 29194

This theorem is used by: tgaaddcpbl2 29232

W3C validator