Theorem inagcgrasubbtwn 33050

Description: In-angle transport, degenerate branch (ii): the sub-angle ⟨“𝐷𝐸𝑊”⟩ is **flat**, 𝐸 lying between 𝐷 and 𝑊. The witness is built beyond 𝑌 on the ray opposite 𝑌𝑋 with axtgsegcon 28926, and flatcgra 29332 supplies the copied angle from the two betweennesses. The in-angle conjunct goes through inagbtwn 33048, which carries the flatness up to the containing angle, then cgrabtwn 29334 across the congruence and isinagd 29358 on the target. (Contributed by JAG, 12-Aug-2026.)

Hypotheses

Ref	Expression
inagcgrasubbtwn.p	⊢ 𝑃 = (Base‘𝐺)
inagcgrasubbtwn.i	⊢ 𝐼 = (Itv‘𝐺)
inagcgrasubbtwn.k	⊢ 𝐾 = (hlG‘𝐺)
inagcgrasubbtwn.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
inagcgrasubbtwn.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
inagcgrasubbtwn.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
inagcgrasubbtwn.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
inagcgrasubbtwn.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
inagcgrasubbtwn.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
inagcgrasubbtwn.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
inagcgrasubbtwn.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
inagcgrasubbtwn.in	⊢ (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
inagcgrasubbtwn.cgb	⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
inagcgrasubbtwn.1	⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑊))

Assertion

Ref	Expression
inagcgrasubbtwn	⊢ (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))

Distinct variable groups:   𝜑,𝑣   𝑣,𝑃   𝑣,𝐼   𝑣,𝐾   𝑣,𝐺   𝑣,𝐷   𝑣,𝐸   𝑣,𝐹   𝑣,𝑊   𝑣,𝑋   𝑣,𝑌   𝑣,𝑍

Proof of Theorem inagcgrasubbtwn

Step	Hyp	Ref	Expression
1		inagcgrasubbtwn.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		inagcgrasubbtwn.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
3		inagcgrasubbtwn.k	. . . 4 ⊢ 𝐾 = (hlG‘𝐺)
4		simplr 781	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣 ∈ 𝑃)
5		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
6		inagcgrasubbtwn.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
7	5, 6	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ∈ 𝑃)
8		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
9		inagcgrasubbtwn.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
10	8, 9	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
11		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
12		inagcgrasubbtwn.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
13	11, 12	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑍 ∈ 𝑃)
14		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
15		inagcgrasubbtwn.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
16	14, 15	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiG)
17		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
18	17, 9	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
19		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
20		inagcgrasubbtwn.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
21		inagcgrasubbtwn.e	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
22		inagcgrasubbtwn.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
23		inagcgrasubbtwn.cgb	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
24	1, 2, 3, 15, 20, 21, 22, 6, 9, 12, 23	cgrane3 29321	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
25	24	necomd 3010	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
26	19, 25	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ≠ 𝑌)
27		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
28	1, 2, 3, 15, 20, 21, 22, 6, 9, 12, 23	cgrane4 29322	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
29	28	necomd 3010	. . . . 5 ⊢ (𝜑 → 𝑍 ≠ 𝑌)
30	27, 29	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑍 ≠ 𝑌)
31		eqid 2760	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
32		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
33	32, 15	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiG)
34		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
35	34, 6	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ∈ 𝑃)
36		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
37	36, 9	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
38		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
39	38, 9	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
40		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣 ∈ 𝑃)
41		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌)))
42	41	simprd 501	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))
43	42	eqcomd 2766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑋(dist‘𝐺)𝑌) = (𝑌(dist‘𝐺)𝑣))
44		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
45	1, 2, 3, 15, 20, 21, 22, 6, 9, 12, 23	cgrane3 29321	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
46	45	necomd 3010	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
47	44, 46	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ≠ 𝑌)
48	1, 31, 2, 33, 35, 37, 39, 40, 43, 47	tgcgrneq 28945	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ≠ 𝑣)
49	48	necomd 3010	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣 ≠ 𝑌)
50		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
51		eqid 2760	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
52		inagcgrasubbtwn.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
53		inagcgrasubbtwn.in	. . . . . . 7 ⊢ (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
54		inagcgrasubbtwn.1	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝑊))
55	1, 2, 3, 15, 20, 21, 22, 52, 53, 54	inagbtwn 33048	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝐷𝐼𝐹))
56	1, 2, 51, 15, 20, 21, 22, 6, 9, 12, 23, 55	cgrabtwn 29334	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑋𝐼𝑍))
57	50, 56	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ (𝑋𝐼𝑍))
58		eqidd 2761	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 = 𝑌)
59	58	orcd 887	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑌 = 𝑌 ∨ 𝑌(𝐾‘𝑌)𝑣))
60	1, 2, 3, 4, 7, 10, 13, 16, 18, 26, 30, 49, 57, 59	isinagd 29358	. . 3 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
61		eqid 2760	. . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺)
62		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
63	62, 15	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiG)
64		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
65	64, 20	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐷 ∈ 𝑃)
66		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
67	66, 21	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐸 ∈ 𝑃)
68		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
69	68, 52	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑊 ∈ 𝑃)
70		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
71	70, 6	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ∈ 𝑃)
72		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
73	72, 9	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
74		simplr 781	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣 ∈ 𝑃)
75		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
76	75, 54	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐸 ∈ (𝐷𝐼𝑊))
77		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌)))
78	77	simpld 500	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ (𝑋𝐼𝑣))
79		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
80	1, 2, 3, 52, 20, 21, 22, 15, 53	inagne1 29361	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
81	79, 80	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐷 ≠ 𝐸)
82		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
83	1, 2, 3, 52, 20, 21, 22, 15, 53	inagne3 29363	. . . . 5 ⊢ (𝜑 → 𝑊 ≠ 𝐸)
84	82, 83	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑊 ≠ 𝐸)
85		simpll 779	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
86	1, 2, 3, 15, 20, 21, 22, 6, 9, 12, 23	cgrane3 29321	. . . . . 6 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
87	86	necomd 3010	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
88	85, 87	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ≠ 𝑌)
89		eqid 2760	. . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺)
90		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
91	90, 15	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝐺 ∈ TarskiG)
92		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
93	92, 6	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ∈ 𝑃)
94		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
95	94, 9	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
96		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
97	96, 9	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ∈ 𝑃)
98		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣 ∈ 𝑃)
99		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌)))
100	99	simprd 501	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))
101	100	eqcomd 2766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑋(dist‘𝐺)𝑌) = (𝑌(dist‘𝐺)𝑣))
102		simpll 779	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝜑)
103	1, 2, 3, 15, 20, 21, 22, 6, 9, 12, 23	cgrane3 29321	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≠ 𝑋)
104	103	necomd 3010	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
105	102, 104	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑋 ≠ 𝑌)
106	1, 89, 2, 91, 93, 95, 97, 98, 101, 105	tgcgrneq 28945	. . . . 5 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑌 ≠ 𝑣)
107	106	necomd 3010	. . . 4 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → 𝑣 ≠ 𝑌)
108	1, 2, 61, 63, 65, 67, 69, 71, 73, 74, 76, 78, 81, 84, 88, 107	flatcgra 29332	. . 3 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩)
109	60, 108	jca 521	. 2 ⊢ (((𝜑 ∧ 𝑣 ∈ 𝑃) ∧ (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌))) → (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
110		eqid 2760	. . 3 ⊢ (dist‘𝐺) = (dist‘𝐺)
111	1, 110, 2, 15, 6, 9, 6, 9	axtgsegcon 28926	. 2 ⊢ (𝜑 → ∃𝑣 ∈ 𝑃 (𝑌 ∈ (𝑋𝐼𝑣) ∧ (𝑌(dist‘𝐺)𝑣) = (𝑋(dist‘𝐺)𝑌)))
112	109, 111	reximddv3 3179	1 ⊢ (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∃wrex 3086   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  ⟨“cs3 14993  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  Itvcitv 28895  hlGchlg 29063  cgrAccgra 29314  inAcinag 29354

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 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  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 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-er 8717  df-map 8849  df-pm 8850  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-n0 12607  df-xnn0 12680  df-z 12694  df-uz 12966  df-fz 13640  df-fzo 13789  df-hash 14475  df-word 14659  df-concat 14716  df-s1 14743  df-s2 14999  df-s3 15000  df-trkgc 28910  df-trkgb 28911  df-trkgcb 28912  df-trkg 28915  df-cgrg 28974  df-hlg 29064  df-cgra 29315  df-inag 29356

This theorem is used by:  cgrainag  33060