Theorem cgrainag 33090

Description: Target-side in-angle transport: if 𝑊 lies inside angle ⟨“𝐷𝐸𝐹”⟩ and that angle is congruent to ⟨“𝑋𝑌𝑍”⟩, then some point inside ⟨“𝑋𝑌𝑍”⟩ makes ⟨“𝑋𝑌𝑣”⟩ congruent to the sub-angle ⟨“𝐷𝐸𝑊”⟩. The geometric core of leawittrg 33091. (Contributed by JAG, 12-Aug-2026.)

Hypotheses

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

Assertion

Ref	Expression
cgrainag	⊢ (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 cgrainag

Step	Hyp	Ref	Expression
1		cgrainag.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
2		cgrainag.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
3		cgrainag.k	. . . 4 ⊢ 𝐾 = (hlG‘𝐺)
4		cgrainag.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐺 ∈ TarskiG)
6	5	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝐺 ∈ TarskiG)
7		cgrainag.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
8	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐷 ∈ 𝑃)
9	8	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝐷 ∈ 𝑃)
10		cgrainag.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
11	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐸 ∈ 𝑃)
12	11	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝐸 ∈ 𝑃)
13		cgrainag.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
14	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐹 ∈ 𝑃)
15	14	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝐹 ∈ 𝑃)
16		cgrainag.w	. . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
17	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊 ∈ 𝑃)
18	17	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝑊 ∈ 𝑃)
19		cgrainag.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
20	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑋 ∈ 𝑃)
21	20	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝑋 ∈ 𝑃)
22		cgrainag.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
23	22	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑌 ∈ 𝑃)
24	23	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝑌 ∈ 𝑃)
25		cgrainag.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
26	25	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑍 ∈ 𝑃)
27	26	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝑍 ∈ 𝑃)
28		cgrainag.cgb	. . . . . 6 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
29	28	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
30	29	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
31		simpr 490	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → 𝐷(𝐾‘𝐸)𝑊)
32	1, 2, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 31	inagcgrasubhl 33079	. . 3 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐷(𝐾‘𝐸)𝑊) → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
33	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐺 ∈ TarskiG)
34	33	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝐺 ∈ TarskiG)
35	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐷 ∈ 𝑃)
36	35	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝐷 ∈ 𝑃)
37	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐸 ∈ 𝑃)
38	37	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝐸 ∈ 𝑃)
39	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐹 ∈ 𝑃)
40	39	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝐹 ∈ 𝑃)
41	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊 ∈ 𝑃)
42	41	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝑊 ∈ 𝑃)
43	19	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑋 ∈ 𝑃)
44	43	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝑋 ∈ 𝑃)
45	22	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑌 ∈ 𝑃)
46	45	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝑌 ∈ 𝑃)
47	25	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑍 ∈ 𝑃)
48	47	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝑍 ∈ 𝑃)
49		cgrainag.in	. . . . . 6 ⊢ (𝜑 → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
50	49	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
51	50	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
52	28	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
53	52	adantr 486	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
54		simpr 490	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝐸 ∈ (𝐷𝐼𝑊))
55	1, 2, 3, 34, 36, 38, 40, 42, 44, 46, 48, 51, 53, 54	inagcgrasubbtwn 33080	. . 3 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
56		simpl 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → (𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)))
57	16	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊 ∈ 𝑃)
58	56, 57	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → 𝑊 ∈ 𝑃)
59		simpl 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → (𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)))
60	7	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐷 ∈ 𝑃)
61	59, 60	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → 𝐷 ∈ 𝑃)
62		simpl 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → (𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)))
63	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐸 ∈ 𝑃)
64	62, 63	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → 𝐸 ∈ 𝑃)
65		simpl 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → (𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)))
66	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐺 ∈ TarskiG)
67	65, 66	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → 𝐺 ∈ TarskiG)
68		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → 𝑊(𝐾‘𝐸)𝐷)
69	1, 2, 3, 58, 61, 64, 67, 68	hlcomd 29100	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → 𝐷(𝐾‘𝐸)𝑊)
70	69	orcd 887	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝑊(𝐾‘𝐸)𝐷) → (𝐷(𝐾‘𝐸)𝑊 ∨ 𝐸 ∈ (𝐷𝐼𝑊)))
71		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → 𝐸 ∈ (𝐷𝐼𝑊))
72	71	olcd 888	. . . 4 ⊢ (((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) ∧ 𝐸 ∈ (𝐷𝐼𝑊)) → (𝐷(𝐾‘𝐸)𝑊 ∨ 𝐸 ∈ (𝐷𝐼𝑊)))
73	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐷 ∈ 𝑃)
74	10	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐸 ∈ 𝑃)
75	16	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊 ∈ 𝑃)
76	4	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐺 ∈ TarskiG)
77	7	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐷 ∈ 𝑃)
78		eqid 2763	. . . . 5 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
79		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸))
80	1, 2, 3, 16, 7, 10, 13, 4, 49	inagne1 29391	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
81	80	neneqd 2964	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝐷 = 𝐸)
82	81	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ¬ 𝐷 = 𝐸)
83		orel2 904	. . . . . . 7 ⊢ (¬ 𝐷 = 𝐸 → ((𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸) → 𝑊 ∈ (𝐷(LineG‘𝐺)𝐸)))
84	82, 83	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ((𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸) → 𝑊 ∈ (𝐷(LineG‘𝐺)𝐸)))
85	79, 84	mpd 16	. . . . 5 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊 ∈ (𝐷(LineG‘𝐺)𝐸))
86	1, 2, 3, 73, 74, 75, 76, 77, 78, 85	lnhl 29111	. . . 4 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → (𝑊(𝐾‘𝐸)𝐷 ∨ 𝐸 ∈ (𝐷𝐼𝑊)))
87	70, 72, 86	mpjaodan 973	. . 3 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → (𝐷(𝐾‘𝐸)𝑊 ∨ 𝐸 ∈ (𝐷𝐼𝑊)))
88	32, 55, 87	mpjaodan 973	. 2 ⊢ ((𝜑 ∧ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
89	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐺 ∈ TarskiG)
90	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐷 ∈ 𝑃)
91	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐸 ∈ 𝑃)
92	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝐹 ∈ 𝑃)
93	16	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊 ∈ 𝑃)
94	19	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑋 ∈ 𝑃)
95	22	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑌 ∈ 𝑃)
96	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑍 ∈ 𝑃)
97	28	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑍”⟩)
98		eqid 2763	. . 3 ⊢ (LineG‘𝐺) = (LineG‘𝐺)
99	49	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → 𝑊(inA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
100		simpr 490	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸))
101	1, 2, 3, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100	inagcgratrgn 33088	. 2 ⊢ ((𝜑 ∧ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)) → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))
102		exmid 908	. . 3 ⊢ ((𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸) ∨ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸))
103	102	a1i 11	. 2 ⊢ (𝜑 → ((𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸) ∨ ¬ (𝑊 ∈ (𝐷(LineG‘𝐺)𝐸) ∨ 𝐷 = 𝐸)))
104	88, 101, 103	mpjaodan 973	1 ⊢ (𝜑 → ∃𝑣 ∈ 𝑃 (𝑣(inA‘𝐺)⟨“𝑋𝑌𝑍”⟩ ∧ ⟨“𝐷𝐸𝑊”⟩(cgrA‘𝐺)⟨“𝑋𝑌𝑣”⟩))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  ⟨“cs3 15021  Basecbs 17416  TarskiGcstrkg 28919  Itvcitv 28925  LineGclng 28926  hlGchlg 29093  cgrAccgra 29344  inAcinag 29384

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  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 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-er 8745  df-map 8877  df-pm 8878  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-4 12434  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-fz 13667  df-fzo 13816  df-hash 14502  df-word 14686  df-concat 14743  df-s1 14770  df-s2 15027  df-s3 15028  df-s4 15029  df-trkgc 28940  df-trkgb 28941  df-trkgcb 28942  df-trkgld 28944  df-trkg 28945  df-cgrg 29004  df-ismt 29026  df-leg 29076  df-hlg 29094  df-mir 29155  df-rag 29199  df-perpg 29201  df-hpg 29266  df-mid 29309  df-lmi 29310  df-cgra 29345  df-inag 29386

This theorem is used by:  leawittrg  33091