Theorem tgasa1 28926

Description: Second congruence theorem: ASA. (Angle-Side-Angle): If two pairs of angles of two triangles are equal in measurement, and the included sides are equal in length, then the triangles are congruent. Theorem 11.50 of [Schwabhauser] p. 108. (Contributed by Thierry Arnoux, 15-Aug-2020.)

Hypotheses

Ref	Expression
tgsas.p	⊢ 𝑃 = (Base‘𝐺)
tgsas.m	⊢ − = (dist‘𝐺)
tgsas.i	⊢ 𝐼 = (Itv‘𝐺)
tgsas.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgsas.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgsas.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgsas.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgsas.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgsas.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
tgsas.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
tgasa.l	⊢ 𝐿 = (LineG‘𝐺)
tgasa.1	⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
tgasa.2	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
tgasa.3	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
tgasa.4	⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)

Assertion

Ref	Expression
tgasa1	⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

Proof of Theorem tgasa1

Dummy variables 𝑎 𝑏 𝑓 𝑤 𝑡 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprr 773	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸 − 𝑓) = (𝐵 − 𝐶))
2		tgsas.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
3		tgsas.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		tgasa.l	. . . . 5 ⊢ 𝐿 = (LineG‘𝐺)
5		tgsas.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	5	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐺 ∈ TarskiG)
7		tgsas.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
8	7	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐹 ∈ 𝑃)
9		tgsas.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
10	9	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐷 ∈ 𝑃)
11		tgsas.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
12	11	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐸 ∈ 𝑃)
13		tgsas.m	. . . . . . 7 ⊢ − = (dist‘𝐺)
14		tgsas.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
15		tgsas.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
16		tgsas.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
17		tgasa.3	. . . . . . 7 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
18		tgasa.1	. . . . . . 7 ⊢ (𝜑 → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
19	2, 3, 13, 5, 14, 15, 16, 9, 11, 7, 17, 4, 18	cgrancol 28897	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
20	19	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21		eqid 2736	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
22		simplr 769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ 𝑃)
23	16	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐶 ∈ 𝑃)
24	14	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐴 ∈ 𝑃)
25	15	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐵 ∈ 𝑃)
26	18	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
27	5	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
28	9	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷 ∈ 𝑃)
29	11	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸 ∈ 𝑃)
30	7	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹 ∈ 𝑃)
31	14	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴 ∈ 𝑃)
32	15	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵 ∈ 𝑃)
33	16	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐶 ∈ 𝑃)
34	2, 3, 5, 21, 14, 15, 16, 9, 11, 7, 17	cgracom 28890	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
35	34	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
37	2, 4, 3, 27, 28, 30, 29, 36	colcom 28626	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐸 ∈ (𝐹𝐿𝐷) ∨ 𝐹 = 𝐷))
38	2, 4, 3, 27, 30, 28, 29, 37	colrot1 28627	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
39	2, 3, 13, 27, 28, 29, 30, 31, 32, 33, 35, 4, 38	cgracol 28896	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
40	18	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
41	39, 40	pm2.65da 817	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42		eqid 2736	. . . . . . . . . 10 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
43	17	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓((hlG‘𝐺)‘𝐸)𝐹)
45	2, 3, 21, 6, 24, 25, 23, 10, 12, 8, 43, 22, 44	cgrahl2 28885	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝑓”⟩)
46	2, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17	cgrane1 28880	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	2, 3, 21, 14, 14, 15, 5, 46	hlid 28677	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴)
48	47	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐴((hlG‘𝐺)‘𝐵)𝐴)
49	2, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17	cgrane2 28881	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
50	49	necomd 2987	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
51	2, 3, 21, 16, 14, 15, 5, 50	hlid 28677	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶((hlG‘𝐺)‘𝐵)𝐶)
52	51	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐶((hlG‘𝐺)‘𝐵)𝐶)
53		tgasa.2	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐷 − 𝐸))
54	2, 13, 3, 5, 14, 15, 9, 11, 53	tgcgrcomlr 28548	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
55	54	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐴) = (𝐸 − 𝐷))
56	1	eqcomd 2742	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝐸 − 𝑓))
57	2, 3, 21, 6, 24, 25, 23, 10, 12, 22, 45, 24, 13, 23, 48, 52, 55, 56	cgracgr 28886	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐴 − 𝐶) = (𝐷 − 𝑓))
58	2, 13, 3, 6, 24, 23, 10, 22, 57	tgcgrcomlr 28548	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐶 − 𝐴) = (𝑓 − 𝐷))
59	53	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐴 − 𝐵) = (𝐷 − 𝐸))
60	2, 13, 42, 6, 23, 24, 25, 22, 10, 12, 58, 59, 56	trgcgr 28584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrG‘𝐺)⟨“𝑓𝐷𝐸”⟩)
61	2, 3, 4, 5, 16, 14, 15, 18	ncolne1 28693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
62	61	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐶 ≠ 𝐴)
63	2, 13, 3, 6, 23, 24, 22, 10, 58, 62	tgcgrneq 28551	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ≠ 𝐷)
64	2, 3, 21, 22, 8, 10, 6, 63	hlid 28677	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓((hlG‘𝐺)‘𝐷)𝑓)
65		tgasa.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐹𝐷𝐸”⟩)
66	2, 3, 21, 5, 16, 14, 15, 7, 9, 11, 65	cgrane4 28883	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
67	66	necomd 2987	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
68	2, 3, 21, 11, 14, 9, 5, 67	hlid 28677	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
69	68	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
70	2, 3, 21, 6, 23, 24, 25, 22, 10, 12, 22, 12, 60, 64, 69	iscgrad 28879	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
71	66	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐷 ≠ 𝐸)
72	2, 3, 6, 21, 22, 10, 12, 63, 71	cgraswap 28888	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝑓𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝑓”⟩)
73	2, 3, 6, 21, 23, 24, 25, 22, 10, 12, 70, 12, 10, 22, 72	cgratr 28891	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝑓”⟩)
74	2, 3, 21, 5, 16, 14, 15, 7, 9, 11, 65	cgrane3 28882	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 𝐹)
75	74	necomd 2987	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ≠ 𝐷)
76	2, 3, 5, 21, 7, 9, 11, 75, 66	cgraswap 28888	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
77	2, 3, 5, 21, 16, 14, 15, 7, 9, 11, 65, 11, 9, 7, 76	cgratr 28891	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
78	77	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
79	2, 3, 4, 5, 11, 9, 67	tgelrnln 28698	. . . . . . . . 9 ⊢ (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
80	79	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
82	81	eleq1d 2821	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
83		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
84	83	eleq1d 2821	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
85	82, 84	anbi12d 633	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
86		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
87		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
88		simplr 769	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
89	87, 88	oveq12d 7385	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
90	86, 89	eleq12d 2830	. . . . . . . . . . 11 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
91	90	cbvrexdva 3218	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
92	85, 91	anbi12d 633	. . . . . . . . 9 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣))))
93	92	cbvopabv 5158	. . . . . . . 8 ⊢ {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑢, 𝑣⟩ ∣ ((𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ∧ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣))}
94	2, 3, 4, 5, 11, 9, 67	tglinerflx1 28701	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐷))
95	94	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐸 ∈ (𝐸𝐿𝐷))
96	2, 4, 3, 5, 9, 11, 7, 19	ncolcom 28629	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
97		pm2.45 882	. . . . . . . . . . 11 ⊢ (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
98	96, 97	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
99	98	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
100	2, 3, 21, 22, 8, 12, 6, 44	hlcomd 28672	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐹((hlG‘𝐺)‘𝐸)𝑓)
101	2, 3, 4, 6, 80, 12, 93, 21, 95, 8, 22, 99, 100	hphl 28839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝑓)
102	2, 3, 4, 6, 80, 8, 93, 22, 101	hpgcom 28835	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
103	2, 3, 4, 5, 79, 7, 93, 98	hpgid 28834	. . . . . . . 8 ⊢ (𝜑 → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
104	103	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
105	2, 3, 13, 6, 23, 24, 25, 12, 10, 8, 4, 26, 41, 22, 8, 21, 73, 78, 102, 104	acopyeu 28902	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓((hlG‘𝐺)‘𝐷)𝐹)
106	2, 3, 21, 22, 8, 10, 6, 4, 105	hlln 28675	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
107	2, 3, 4, 5, 7, 9, 75	tglinerflx1 28701	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐹𝐿𝐷))
108	107	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐹 ∈ (𝐹𝐿𝐷))
109	2, 3, 21, 5, 14, 15, 16, 9, 11, 7, 17	cgrane4 28883	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
110	109	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐸 ≠ 𝐹)
111	2, 3, 21, 22, 8, 12, 6, 4, 44	hlln 28675	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐸))
112	2, 3, 4, 6, 12, 8, 22, 110, 111	lncom 28690	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
113	2, 3, 4, 6, 12, 8, 110	tglinerflx2 28702	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐹 ∈ (𝐸𝐿𝐹))
114	2, 3, 4, 6, 8, 10, 12, 8, 20, 106, 108, 112, 113	tglineinteq 28713	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 = 𝐹)
115	114	oveq2d 7383	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸 − 𝑓) = (𝐸 − 𝐹))
116	1, 115	eqtr3d 2773	. 2 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
117	109	necomd 2987	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
118	2, 3, 21, 11, 15, 16, 5, 7, 13, 117, 49	hlcgrex 28684	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶)))
119	116, 118	r19.29a 3145	1 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061   ∖ cdif 3886   class class class wbr 5085  {copab 5147  ran crn 5632  ‘cfv 6498  (class class class)co 7367  ⟨“cs3 14804  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501  LineGclng 28502  cgrGccgrg 28578  hlGchlg 28668  hpGchpg 28825  cgrAccgra 28875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-er 8643  df-map 8775  df-pm 8776  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-fz 13462  df-fzo 13609  df-hash 14293  df-word 14476  df-concat 14533  df-s1 14559  df-s2 14810  df-s3 14811  df-trkgc 28516  df-trkgb 28517  df-trkgcb 28518  df-trkgld 28520  df-trkg 28521  df-cgrg 28579  df-leg 28651  df-hlg 28669  df-mir 28721  df-rag 28762  df-perpg 28764  df-hpg 28826  df-mid 28842  df-lmi 28843  df-cgra 28876

This theorem is referenced by:  tgasa  28927