Theorem tgasa1 28914

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 28885	. . . . . 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 2737	. . . . . 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 28878	. . . . . . . . . 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 28614	. . . . . . . . . 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 28615	. . . . . . . . 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 28884	. . . . . . . 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 2737	. . . . . . . . . 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 28873	. . . . . . . . . . . 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 28868	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	2, 3, 21, 14, 14, 15, 5, 46	hlid 28665	. . . . . . . . . . . . 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 28869	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
50	49	necomd 2988	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
51	2, 3, 21, 16, 14, 15, 5, 50	hlid 28665	. . . . . . . . . . . . 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 28536	. . . . . . . . . . . . 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 2743	. . . . . . . . . . . 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 28874	. . . . . . . . . . 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 28536	. . . . . . . . . 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 28572	. . . . . . . . 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 28681	. . . . . . . . . . . 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 28539	. . . . . . . . . 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 28665	. . . . . . . . 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 28871	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
67	66	necomd 2988	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
68	2, 3, 21, 11, 14, 9, 5, 67	hlid 28665	. . . . . . . . . 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 28867	. . . . . . . 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 28876	. . . . . . . 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 28879	. . . . . . 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 28870	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 𝐹)
75	74	necomd 2988	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ≠ 𝐷)
76	2, 3, 5, 21, 7, 9, 11, 75, 66	cgraswap 28876	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
77	2, 3, 5, 21, 16, 14, 15, 7, 9, 11, 65, 11, 9, 7, 76	cgratr 28879	. . . . . . . 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 28686	. . . . . . . . 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 2822	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
83		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
84	83	eleq1d 2822	. . . . . . . . . . 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 7376	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
90	86, 89	eleq12d 2831	. . . . . . . . . . 11 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
91	90	cbvrexdva 3219	. . . . . . . . . 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 5159	. . . . . . . 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 28689	. . . . . . . . . 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 28617	. . . . . . . . . . 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 28660	. . . . . . . . 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 28827	. . . . . . . 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 28823	. . . . . . 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 28822	. . . . . . . 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 28890	. . . . . 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 28663	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
107	2, 3, 4, 5, 7, 9, 75	tglinerflx1 28689	. . . . . 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 28871	. . . . . . 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 28663	. . . . . 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 28678	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
113	2, 3, 4, 6, 12, 8, 110	tglinerflx2 28690	. . . . 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 28701	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 = 𝐹)
115	114	oveq2d 7374	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸 − 𝑓) = (𝐸 − 𝐹))
116	1, 115	eqtr3d 2774	. 2 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
117	109	necomd 2988	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
118	2, 3, 21, 11, 15, 16, 5, 7, 13, 117, 49	hlcgrex 28672	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶)))
119	116, 118	r19.29a 3146	1 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062   ∖ cdif 3887   class class class wbr 5086  {copab 5148  ran crn 5623  ‘cfv 6490  (class class class)co 7358  ⟨“cs3 14766  Basecbs 17137  distcds 17187  TarskiGcstrkg 28483  Itvcitv 28489  LineGclng 28490  cgrGccgrg 28566  hlGchlg 28656  hpGchpg 28813  cgrAccgra 28863

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 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-cnex 11083  ax-resscn 11084  ax-1cn 11085  ax-icn 11086  ax-addcl 11087  ax-addrcl 11088  ax-mulcl 11089  ax-mulrcl 11090  ax-mulcom 11091  ax-addass 11092  ax-mulass 11093  ax-distr 11094  ax-i2m1 11095  ax-1ne0 11096  ax-1rid 11097  ax-rnegex 11098  ax-rrecex 11099  ax-cnre 11100  ax-pre-lttri 11101  ax-pre-lttrn 11102  ax-pre-ltadd 11103  ax-pre-mulgt0 11104

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-oadd 8400  df-er 8634  df-map 8766  df-pm 8767  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-dju 9814  df-card 9852  df-pnf 11169  df-mnf 11170  df-xr 11171  df-ltxr 11172  df-le 11173  df-sub 11367  df-neg 11368  df-nn 12147  df-2 12209  df-3 12210  df-n0 12403  df-xnn0 12476  df-z 12490  df-uz 12753  df-fz 13425  df-fzo 13572  df-hash 14255  df-word 14438  df-concat 14495  df-s1 14521  df-s2 14772  df-s3 14773  df-trkgc 28504  df-trkgb 28505  df-trkgcb 28506  df-trkgld 28508  df-trkg 28509  df-cgrg 28567  df-leg 28639  df-hlg 28657  df-mir 28709  df-rag 28750  df-perpg 28752  df-hpg 28814  df-mid 28830  df-lmi 28831  df-cgra 28864

This theorem is referenced by:  tgasa  28915