Theorem tgasa1 29403

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 785	. . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 739	. . . . 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 29337	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
20	19	ad2antrr 739	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21		eqid 2761	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
22		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ 𝑃)
23	16	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐶 ∈ 𝑃)
24	14	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐴 ∈ 𝑃)
25	15	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐵 ∈ 𝑃)
26	18	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
27	5	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
28	9	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷 ∈ 𝑃)
29	11	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸 ∈ 𝑃)
30	7	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹 ∈ 𝑃)
31	14	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴 ∈ 𝑃)
32	15	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵 ∈ 𝑃)
33	16	ad3antrrr 743	. . . . . . . . 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 29329	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
35	34	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
36		simpr 490	. . . . . . . . . . 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 29021	. . . . . . . . . 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 29022	. . . . . . . . 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 29336	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
40	18	ad3antrrr 743	. . . . . . . 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 829	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42		eqid 2761	. . . . . . . . . 10 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
43	17	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44		simprl 783	. . . . . . . . . . . . 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 29324	. . . . . . . . . . . 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 29319	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	2, 3, 21, 14, 14, 15, 5, 46	hlid 29075	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴)
48	47	ad2antrr 739	. . . . . . . . . . . 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 29320	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
50	49	necomd 3011	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
51	2, 3, 21, 16, 14, 15, 5, 50	hlid 29075	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶((hlG‘𝐺)‘𝐵)𝐶)
52	51	ad2antrr 739	. . . . . . . . . . . 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 28942	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
55	54	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐴) = (𝐸 − 𝐷))
56	1	eqcomd 2767	. . . . . . . . . . . 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 29325	. . . . . . . . . . 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 28942	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐶 − 𝐴) = (𝑓 − 𝐷))
59	53	ad2antrr 739	. . . . . . . . . 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 28979	. . . . . . . . 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 29093	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
62	61	ad2antrr 739	. . . . . . . . . . 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 28945	. . . . . . . . . 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 29075	. . . . . . . . 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 29322	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
67	66	necomd 3011	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
68	2, 3, 21, 11, 14, 9, 5, 67	hlid 29075	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
69	68	ad2antrr 739	. . . . . . . . 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 29318	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
71	66	ad2antrr 739	. . . . . . . . 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 29327	. . . . . . . 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 29330	. . . . . . 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 29321	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 𝐹)
75	74	necomd 3011	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ≠ 𝐷)
76	2, 3, 5, 21, 7, 9, 11, 75, 66	cgraswap 29327	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
77	2, 3, 5, 21, 16, 14, 15, 7, 9, 11, 65, 11, 9, 7, 76	cgratr 29330	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
78	77	ad2antrr 739	. . . . . . 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 29098	. . . . . . . . 9 ⊢ (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
80	79	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸𝐿𝐷) ∈ ran 𝐿)
81		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑎 = 𝑢)
82	81	eleq1d 2846	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
83		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
84	83	eleq1d 2846	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
85	82, 84	anbi12d 644	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → ((𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷))) ↔ (𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ∧ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷)))))
86		simpr 490	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑡 = 𝑤)
87		simpll 779	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
88		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
89	87, 88	oveq12d 7438	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
90	86, 89	eleq12d 2855	. . . . . . . . . . 11 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑤 ∈ (𝑢𝐼𝑣)))
91	90	cbvrexdva 3244	. . . . . . . . . 10 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (∃𝑡 ∈ (𝐸𝐿𝐷)𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑤 ∈ (𝐸𝐿𝐷)𝑤 ∈ (𝑢𝐼𝑣)))
92	85, 91	anbi12d 644	. . . . . . . . 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 5178	. . . . . . . 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 29101	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐷))
95	94	ad2antrr 739	. . . . . . . . 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 29024	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
97		pm2.45 895	. . . . . . . . . . 11 ⊢ (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
98	96, 97	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
99	98	ad2antrr 739	. . . . . . . . 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 29070	. . . . . . . . 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 29249	. . . . . . . 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 29245	. . . . . . 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 29244	. . . . . . . 8 ⊢ (𝜑 → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
104	103	ad2antrr 739	. . . . . . 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 29342	. . . . . 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 29073	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
107	2, 3, 4, 5, 7, 9, 75	tglinerflx1 29101	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐹𝐿𝐷))
108	107	ad2antrr 739	. . . . 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 29322	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
110	109	ad2antrr 739	. . . . . 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 29073	. . . . . 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 29090	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
113	2, 3, 4, 6, 12, 8, 110	tglinerflx2 29102	. . . . 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 29114	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 = 𝐹)
115	114	oveq2d 7436	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸 − 𝑓) = (𝐸 − 𝐹))
116	1, 115	eqtr3d 2798	. 2 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
117	109	necomd 3011	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
118	2, 3, 21, 11, 15, 16, 5, 7, 13, 117, 49	hlcgrex 29082	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶)))
119	116, 118	r19.29a 3171	1 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087   ∖ cdif 3896   class class class wbr 5103  {copab 5167  ran crn 5652  ‘cfv 6538  (class class class)co 7420  ⟨“cs3 14993  Basecbs 17387  distcds 17437  TarskiGcstrkg 28889  Itvcitv 28895  LineGclng 28896  cgrGccgrg 28973  hlGchlg 29063  hpGchpg 29235  cgrAccgra 29314

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  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 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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-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-trkgc 28910  df-trkgb 28911  df-trkgcb 28912  df-trkgld 28914  df-trkg 28915  df-cgrg 28974  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

This theorem is used by:  tgasa  29404