Theorem tgaaddcpbl 29206

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 2771	. . . . . . 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 2765	. . . . . . . 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 2765	. . . . . . . . 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 28937	. . . . . . . . 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 28957	. . . . . . . . . . . 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 28954	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑌𝐿𝑆) ∈ ran 𝐿)
37		tgaaddcpbl.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋𝑂𝑍)
38	4, 24, 5, 35, 31, 36, 7, 10, 15, 37	oppne1 29073	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑆))
39		nelne2 3058	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑋 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑋)
40	34, 38, 39	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
41	40	necomd 3015	. . . . . . . . . 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 29074	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑌𝐿𝑆))
44		nelne2 3058	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ (𝑌𝐿𝑆) ∧ ¬ 𝑍 ∈ (𝑌𝐿𝑆)) → 𝑌 ≠ 𝑍)
45	34, 43, 44	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
46	45	necomd 3015	. . . . . . . . . 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 28957	. . . . . . . . . . . 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 29071	. . . . . . . . . . . . . 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 3058	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ (𝑉𝐿𝑇) ∧ ¬ 𝑈 ∈ (𝑉𝐿𝑇)) → 𝑉 ≠ 𝑈)
58	50, 56, 57	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
59	58	necomd 3015	. . . . . . . . . 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 2771	. . . . . . . . . . 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 28803	. . . . . . . . . 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 3015	. . . . . . . . 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 29186	. . . . . . . 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 2765	. . . . . . . . . . 11 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺)
71		eqid 2765	. . . . . . . . . . 11 ⊢ ((pInvG‘𝐺)‘𝑇) = ((pInvG‘𝐺)‘𝑇)
72	4, 24, 5, 31, 70, 7, 48, 71, 18	mircl 28989	. . . . . . . . . 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 28959	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → (𝑌𝐿𝑍) = (𝑍𝐿𝑌))
82	79, 81	eleqtrd 2867	. . . . . . . . . . . . 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 28947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 ∈ (𝑌𝐿𝑍)) → 𝑍 ∈ (𝑌𝐿𝑆))
85	43, 84	mtand 828	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ 𝑆 ∈ (𝑌𝐿𝑍))
86	45	neneqd 2965	. . . . . . . . . . 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 28990	. . . . . . . . . . . 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 28954	. . . . . . . . . . . . . 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 28958	. . . . . . . . . . . . . 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 28878	. . . . . . . . . . . . . . 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 28879	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑇 ∈ (𝑉𝐿(((pInvG‘𝐺)‘𝑇)‘𝑈)) ∨ 𝑉 = (((pInvG‘𝐺)‘𝑇)‘𝑈))) → ((((pInvG‘𝐺)‘𝑇)‘𝑈) ∈ (𝑉𝐿𝑇) ∨ 𝑉 = 𝑇))
103	49	neneqd 2965	. . . . . . . . . . . . . . 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 29004	. . . . . . . . . . . 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 2866	. . . . . . . . . . 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 5126	. . . . . . . . . 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 5126	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩(cgrA‘𝐺)⟨“𝑈𝑉𝑇”⟩)
116	4, 5, 7, 6, 10, 12, 32, 18, 20, 48, 115	cgracom 29184	. . . . . . . . . . . . 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 29193	. . . . . . . . . . 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 29187	. . . . . . . . . 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 29184	. . . . . . . . 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 28959	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑉𝐿𝑇) = (𝑇𝐿𝑉))
122	121	fveq2d 6889	. . . . . . . . . . 11 ⊢ (𝜑 → ((hpG‘𝐺)‘(𝑉𝐿𝑇)) = ((hpG‘𝐺)‘(𝑇𝐿𝑉)))
123	18, 56	eldifd 3917	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ (𝑃 ∖ (𝑉𝐿𝑇)))
124	4, 5, 70, 71, 52, 7, 94, 96, 123, 31	oppmir 29087	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑈𝑄(((pInvG‘𝐺)‘𝑇)‘𝑈))
125	4, 24, 5, 52, 31, 94, 7, 18, 72, 124	oppcom 29076	. . . . . . . . . . . 12 ⊢ (𝜑 → (((pInvG‘𝐺)‘𝑇)‘𝑈)𝑄𝑈)
126	4, 24, 5, 52, 31, 94, 7, 18, 53, 51	oppcom 29076	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑊𝑄𝑈)
127	4, 5, 31, 52, 7, 94, 53, 72, 18, 126	lnopp2hpgb 29096	. . . . . . . . . . . 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 5126	. . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . . . . 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 28803	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝜑 ∧ 𝑌 ∈ (𝑋𝐼𝑍)) ∧ 𝑢 ∈ 𝑃) ∧ 𝑢((hlG‘𝐺)‘𝑉)𝑈) ∧ (𝑉(dist‘𝐺)𝑢) = (𝑌(dist‘𝐺)𝑋)) → 𝑉 ≠ 𝑢)
146	145	necomd 3015	. . . . . . . . . . . . . . . . . . . . 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 28948	. . . . . . . . . . . . . . . . . . 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 28954	. . . . . . . . . . . . . . . . . . . 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 28927	. . . . . . . . . . . . . . . . . . . . . 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 28930	. . . . . . . . . . . . . . . . . . . . 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 28958	. . . . . . . . . . . . . . . . . . . 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 28960	. . . . . . . . . . . . . . . . . . 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 2867	. . . . . . . . . . . . . . . . . 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 28947	. . . . . . . . . . . . . . . . 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 2965	. . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . . . . . . . . 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 2867	. . . . . . . . . . . . . . . . . . . . . 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 28786	. . . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . 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 3051	. . . . . . . . . . . . . . . . . . 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 28954	. . . . . . . . . . . . . . . . 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 28957	. . . . . . . . . . . . . . . . . 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 3057	. . . . . . . . . . . . . . . . . 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 28943	. . . . . . . . . . . . . . . . . 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 4153	. . . . . . . . . . . . . . . . 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 28958	. . . . . . . . . . . . . . . . . 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 4153	. . . . . . . . . . . . . . . . 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 28967	. . . . . . . . . . . . . . . 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 29072	. . . . . . . . . . . . . 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 29086	. . . . . . . . . . . . 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 29076	. . . . . . . . . . . 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 29096	. . . . . . . . . . 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 5126	. . . . . . . . 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 29196	. . . . . . . 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 29179	. . . . . . 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 5126	. . . . . 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 28784	. . . . 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 3175	. . . 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 28939	. . . 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 3175	. 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 29205	. 2 ⊢ ((𝜑 ∧ ¬ 𝑌 ∈ (𝑋𝐼𝑍)) → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
228	210, 227	pm2.61dan 825	1 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091   ∖ cdif 3903   class class class wbr 5111  {copab 5175  ran crn 5664  ‘cfv 6540  (class class class)co 7419  ⟨“cs3 14903  Basecbs 17291  distcds 17341  TarskiGcstrkg 28747  Itvcitv 28753  LineGclng 28754  hlGchlg 28920  pInvGcmir 28980  hpGchpg 29090  cgrAccgra 29169

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-er 8700  df-map 8832  df-pm 8833  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-dju 9903  df-card 9941  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-3 12319  df-n0 12520  df-xnn0 12593  df-z 12607  df-uz 12879  df-fz 13552  df-fzo 13700  df-hash 14385  df-word 14569  df-concat 14626  df-s1 14653  df-s2 14909  df-s3 14910  df-trkgc 28768  df-trkgb 28769  df-trkgcb 28770  df-trkgld 28772  df-trkg 28773  df-cgrg 28831  df-leg 28903  df-hlg 28921  df-mir 28981  df-rag 29025  df-perpg 29027  df-hpg 29091  df-mid 29134  df-lmi 29135  df-cgra 29170

This theorem is used by: (None)