Theorem tgasa1 28785

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 772	. . 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 726	. . . . 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 726	. . . . 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 726	. . . . 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 726	. . . . 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 28756	. . . . . 6 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
20	19	ad2antrr 726	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐹 ∈ (𝐷𝐿𝐸) ∨ 𝐷 = 𝐸))
21		eqid 2729	. . . . . 6 ⊢ (hlG‘𝐺) = (hlG‘𝐺)
22		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ 𝑃)
23	16	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐶 ∈ 𝑃)
24	14	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐴 ∈ 𝑃)
25	15	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝐵 ∈ 𝑃)
26	18	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
27	5	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐺 ∈ TarskiG)
28	9	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐷 ∈ 𝑃)
29	11	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐸 ∈ 𝑃)
30	7	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐹 ∈ 𝑃)
31	14	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐴 ∈ 𝑃)
32	15	ad3antrrr 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → 𝐵 ∈ 𝑃)
33	16	ad3antrrr 730	. . . . . . . . 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 28749	. . . . . . . . . 10 ⊢ (𝜑 → ⟨“𝐷𝐸𝐹”⟩(cgrA‘𝐺)⟨“𝐴𝐵𝐶”⟩)
35	34	ad3antrrr 730	. . . . . . . . 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 28485	. . . . . . . . . 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 28486	. . . . . . . . 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 28755	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) ∧ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹)) → (𝐶 ∈ (𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))
40	18	ad3antrrr 730	. . . . . . . 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 816	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ¬ (𝐸 ∈ (𝐷𝐿𝐹) ∨ 𝐷 = 𝐹))
42		eqid 2729	. . . . . . . . . 10 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
43	17	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
44		simprl 770	. . . . . . . . . . . . 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 28744	. . . . . . . . . . . 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 28739	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
47	2, 3, 21, 14, 14, 15, 5, 46	hlid 28536	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴((hlG‘𝐺)‘𝐵)𝐴)
48	47	ad2antrr 726	. . . . . . . . . . . 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 28740	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
50	49	necomd 2980	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
51	2, 3, 21, 16, 14, 15, 5, 50	hlid 28536	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶((hlG‘𝐺)‘𝐵)𝐶)
52	51	ad2antrr 726	. . . . . . . . . . . 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 28407	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐸 − 𝐷))
55	54	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐴) = (𝐸 − 𝐷))
56	1	eqcomd 2735	. . . . . . . . . . . 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 28745	. . . . . . . . . . 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 28407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐶 − 𝐴) = (𝑓 − 𝐷))
59	53	ad2antrr 726	. . . . . . . . . 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 28443	. . . . . . . . 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 28552	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
62	61	ad2antrr 726	. . . . . . . . . . 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 28410	. . . . . . . . . 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 28536	. . . . . . . . 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 28742	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
67	66	necomd 2980	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ≠ 𝐷)
68	2, 3, 21, 11, 14, 9, 5, 67	hlid 28536	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸((hlG‘𝐺)‘𝐷)𝐸)
69	68	ad2antrr 726	. . . . . . . . 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 28738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝑓𝐷𝐸”⟩)
71	66	ad2antrr 726	. . . . . . . . 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 28747	. . . . . . . 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 28750	. . . . . . 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 28741	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 𝐹)
75	74	necomd 2980	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ≠ 𝐷)
76	2, 3, 5, 21, 7, 9, 11, 75, 66	cgraswap 28747	. . . . . . . . 9 ⊢ (𝜑 → ⟨“𝐹𝐷𝐸”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
77	2, 3, 5, 21, 16, 14, 15, 7, 9, 11, 65, 11, 9, 7, 76	cgratr 28750	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝐶𝐴𝐵”⟩(cgrA‘𝐺)⟨“𝐸𝐷𝐹”⟩)
78	77	ad2antrr 726	. . . . . . 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 28557	. . . . . . . . 9 ⊢ (𝜑 → (𝐸𝐿𝐷) ∈ ran 𝐿)
80	79	ad2antrr 726	. . . . . . . 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 2813	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑎 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑢 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
83		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → 𝑏 = 𝑣)
84	83	eleq1d 2813	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) → (𝑏 ∈ (𝑃 ∖ (𝐸𝐿𝐷)) ↔ 𝑣 ∈ (𝑃 ∖ (𝐸𝐿𝐷))))
85	82, 84	anbi12d 632	. . . . . . . . . 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 766	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑎 = 𝑢)
88		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → 𝑏 = 𝑣)
89	87, 88	oveq12d 7405	. . . . . . . . . . . 12 ⊢ (((𝑎 = 𝑢 ∧ 𝑏 = 𝑣) ∧ 𝑡 = 𝑤) → (𝑎𝐼𝑏) = (𝑢𝐼𝑣))
90	86, 89	eleq12d 2822	. . . . . . . . . . 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 632	. . . . . . . . 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 5180	. . . . . . . 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 28560	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (𝐸𝐿𝐷))
95	94	ad2antrr 726	. . . . . . . . 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 28488	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷))
97		pm2.45 881	. . . . . . . . . . 11 ⊢ (¬ (𝐹 ∈ (𝐸𝐿𝐷) ∨ 𝐸 = 𝐷) → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
98	96, 97	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐸𝐿𝐷))
99	98	ad2antrr 726	. . . . . . . . 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 28531	. . . . . . . . 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 28698	. . . . . . . 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 28694	. . . . . . 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 28693	. . . . . . . 8 ⊢ (𝜑 → 𝐹((hpG‘𝐺)‘(𝐸𝐿𝐷))𝐹)
104	103	ad2antrr 726	. . . . . . 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 28761	. . . . . 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 28534	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐹𝐿𝐷))
107	2, 3, 4, 5, 7, 9, 75	tglinerflx1 28560	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐹𝐿𝐷))
108	107	ad2antrr 726	. . . . 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 28742	. . . . . . 7 ⊢ (𝜑 → 𝐸 ≠ 𝐹)
110	109	ad2antrr 726	. . . . . 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 28534	. . . . . 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 28549	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 ∈ (𝐸𝐿𝐹))
113	2, 3, 4, 6, 12, 8, 110	tglinerflx2 28561	. . . . 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 28572	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → 𝑓 = 𝐹)
115	114	oveq2d 7403	. . 3 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐸 − 𝑓) = (𝐸 − 𝐹))
116	1, 115	eqtr3d 2766	. 2 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝑃) ∧ (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶))) → (𝐵 − 𝐶) = (𝐸 − 𝐹))
117	109	necomd 2980	. . 3 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
118	2, 3, 21, 11, 15, 16, 5, 7, 13, 117, 49	hlcgrex 28543	. 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑃 (𝑓((hlG‘𝐺)‘𝐸)𝐹 ∧ (𝐸 − 𝑓) = (𝐵 − 𝐶)))
119	116, 118	r19.29a 3141	1 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐸 − 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∃wrex 3053   ∖ cdif 3911   class class class wbr 5107  {copab 5169  ran crn 5639  ‘cfv 6511  (class class class)co 7387  ⟨“cs3 14808  Basecbs 17179  distcds 17229  TarskiGcstrkg 28354  Itvcitv 28360  LineGclng 28361  cgrGccgrg 28437  hlGchlg 28527  hpGchpg 28684  cgrAccgra 28734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-oadd 8438  df-er 8671  df-map 8801  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-dju 9854  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-n0 12443  df-xnn0 12516  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616  df-hash 14296  df-word 14479  df-concat 14536  df-s1 14561  df-s2 14814  df-s3 14815  df-trkgc 28375  df-trkgb 28376  df-trkgcb 28377  df-trkgld 28379  df-trkg 28380  df-cgrg 28438  df-leg 28510  df-hlg 28528  df-mir 28580  df-rag 28621  df-perpg 28623  df-hpg 28685  df-mid 28701  df-lmi 28702  df-cgra 28735

This theorem is referenced by:  tgasa  28786