Theorem cgrainag 33060

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 33061. (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 33049	. . 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 33050	. . 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 29070	. . . . 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 2760	. . . . 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 29361	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
81	80	neneqd 2960	. . . . . . . 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 29081	. . . 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 2760	. . 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 33058	. 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 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  ⟨“cs3 14993  Basecbs 17387  TarskiGcstrkg 28889  Itvcitv 28895  LineGclng 28896  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-rmo 3365  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-4 12407  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-s4 15001  df-trkgc 28910  df-trkgb 28911  df-trkgcb 28912  df-trkgld 28914  df-trkg 28915  df-cgrg 28974  df-ismt 28996  df-leg 29046  df-hlg 29064  df-mir 29125  df-rag 29169  df-perpg 29171  df-hpg 29236  df-mid 29279  df-lmi 29280  df-cgra 29315  df-inag 29356

This theorem is used by:  leawittrg  33061