Theorem tgasa1 27123

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 769	. . 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 722	. . . . 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 722	. . . . 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 722	. . . . 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 722	. . . . 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 27094	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
20	19	ad2antrr 722	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21		eqid 2738	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
22		simplr 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ 𝑃)
23	16	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐶 ∈ 𝑃)
24	14	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐴 ∈ 𝑃)
25	15	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐵 ∈ 𝑃)
26	18	ad2antrr 722	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
27	5	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
28	9	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷 ∈ 𝑃)
29	11	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸 ∈ 𝑃)
30	7	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹 ∈ 𝑃)
31	14	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴 ∈ 𝑃)
32	15	ad3antrrr 726	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵 ∈ 𝑃)
33	16	ad3antrrr 726	. . . . . . . . 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 27087	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
35	34	ad3antrrr 726	. . . . . . . . 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 26823	. . . . . . . . . 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 26824	. . . . . . . . 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 27093	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
40	18	ad3antrrr 726	. . . . . . . 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 813	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42		eqid 2738	. . . . . . . . . 10 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
43	17	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44		simprl 767	. . . . . . . . . . . . 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 27082	. . . . . . . . . . . 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 27077	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	2, 3, 21, 14, 14, 15, 5, 46	hlid 26874	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴)
48	47	ad2antrr 722	. . . . . . . . . . . 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 27078	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
50	49	necomd 2998	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
51	2, 3, 21, 16, 14, 15, 5, 50	hlid 26874	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶((hlG‘𝐺)‘𝐵)𝐶)
52	51	ad2antrr 722	. . . . . . . . . . . 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 26745	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
55	54	ad2antrr 722	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐴) = (𝐸 − 𝐷))
56	1	eqcomd 2744	. . . . . . . . . . . 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 27083	. . . . . . . . . . 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 26745	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐶 − 𝐴) = (𝑓 − 𝐷))
59	53	ad2antrr 722	. . . . . . . . . 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 26781	. . . . . . . . 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 26890	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
62	61	ad2antrr 722	. . . . . . . . . . 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 26748	. . . . . . . . . 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 26874	. . . . . . . . 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 27080	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
67	66	necomd 2998	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
68	2, 3, 21, 11, 14, 9, 5, 67	hlid 26874	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
69	68	ad2antrr 722	. . . . . . . . 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 27076	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
71	66	ad2antrr 722	. . . . . . . . 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 27085	. . . . . . . 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 27088	. . . . . . 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 27079	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 𝐹)
75	74	necomd 2998	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ≠ 𝐷)
76	2, 3, 5, 21, 7, 9, 11, 75, 66	cgraswap 27085	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
77	2, 3, 5, 21, 16, 14, 15, 7, 9, 11, 65, 11, 9, 7, 76	cgratr 27088	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
78	77	ad2antrr 722	. . . . . . 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 26895	. . . . . . . . 9 ⊢ (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
80	79	ad2antrr 722	. . . . . . . 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 2823	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
83		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
84	83	eleq1d 2823	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
85	82, 84	anbi12d 630	. . . . . . . . . 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 763	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
88		simplr 765	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
89	87, 88	oveq12d 7273	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
90	86, 89	eleq12d 2833	. . . . . . . . . . 11 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
91	90	cbvrexdva 3384	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
92	85, 91	anbi12d 630	. . . . . . . . 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 5143	. . . . . . . 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 26898	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐷))
95	94	ad2antrr 722	. . . . . . . . 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 26826	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
97		pm2.45 878	. . . . . . . . . . 11 ⊢ (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
98	96, 97	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
99	98	ad2antrr 722	. . . . . . . . 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 26869	. . . . . . . . 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 27036	. . . . . . . 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 27032	. . . . . . 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 27031	. . . . . . . 8 ⊢ (𝜑 → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
104	103	ad2antrr 722	. . . . . . 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 27099	. . . . . 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 26872	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
107	2, 3, 4, 5, 7, 9, 75	tglinerflx1 26898	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐹𝐿𝐷))
108	107	ad2antrr 722	. . . . 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 27080	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
110	109	ad2antrr 722	. . . . . 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 26872	. . . . . 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 26887	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
113	2, 3, 4, 6, 12, 8, 110	tglinerflx2 26899	. . . . 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 26910	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 = 𝐹)
115	114	oveq2d 7271	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸 − 𝑓) = (𝐸 − 𝐹))
116	1, 115	eqtr3d 2780	. 2 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
117	109	necomd 2998	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
118	2, 3, 21, 11, 15, 16, 5, 7, 13, 117, 49	hlcgrex 26881	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶)))
119	116, 118	r19.29a 3217	1 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∃wrex 3064   ∖ cdif 3880   class class class wbr 5070  {copab 5132  ran crn 5581  ‘cfv 6418  (class class class)co 7255  ⟨“cs3 14483  Basecbs 16840  distcds 16897  TarskiGcstrkg 26693  Itvcitv 26699  LineGclng 26700  cgrGccgrg 26775  hlGchlg 26865  hpGchpg 27022  cgrAccgra 27072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-n0 12164  df-xnn0 12236  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-concat 14202  df-s1 14229  df-s2 14489  df-s3 14490  df-trkgc 26713  df-trkgb 26714  df-trkgcb 26715  df-trkgld 26717  df-trkg 26718  df-cgrg 26776  df-leg 26848  df-hlg 26866  df-mir 26918  df-rag 26959  df-perpg 26961  df-hpg 27023  df-mid 27039  df-lmi 27040  df-cgra 27073

This theorem is referenced by:  tgasa  27124