Theorem cgrahlb 33077

Description: Two null angles are congruent: if 𝐴 and 𝐶 lie on one half-line from 𝐵, and 𝐷 and 𝐹 on one half-line from 𝐸, then angle ⟨“𝐴𝐵𝐶”⟩ is congruent to ⟨“𝐷𝐸𝐹”⟩. This is GeoCoq l11_21_b ; set.mm has only the converse, cgrahl 29365 (GeoCoq l11_21_a ). Proved by copying both arms onto the target rays with hlcgrex 29112 and closing the third side by segment subtraction (tgcgrsub2 29088), so it needs no non-collinear frame. (Contributed by JAG, 12-Aug-2026.)

Hypotheses

Ref	Expression
cgrahlb.p	⊢ 𝑃 = (Base‘𝐺)
cgrahlb.i	⊢ 𝐼 = (Itv‘𝐺)
cgrahlb.k	⊢ 𝐾 = (hlG‘𝐺)
cgrahlb.m	⊢ − = (dist‘𝐺)
cgrahlb.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
cgrahlb.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
cgrahlb.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
cgrahlb.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
cgrahlb.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
cgrahlb.e	⊢ (𝜑 → 𝐸 ∈ 𝑃)
cgrahlb.f	⊢ (𝜑 → 𝐹 ∈ 𝑃)
cgrahlb.1	⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶)
cgrahlb.2	⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹)

Assertion

Ref	Expression
cgrahlb	⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

Proof of Theorem cgrahlb

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cgrahlb.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		cgrahlb.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
3		cgrahlb.k	. . 3 ⊢ 𝐾 = (hlG‘𝐺)
4		simplll 787	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝜑)
5		cgrahlb.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6	4, 5	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐺 ∈ TarskiG)
7		cgrahlb.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
8	4, 7	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐴 ∈ 𝑃)
9		cgrahlb.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
10	4, 9	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐵 ∈ 𝑃)
11		cgrahlb.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
12	4, 11	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐶 ∈ 𝑃)
13		cgrahlb.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
14	4, 13	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐷 ∈ 𝑃)
15		cgrahlb.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃)
16	4, 15	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐸 ∈ 𝑃)
17		cgrahlb.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑃)
18	4, 17	syl 18	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐹 ∈ 𝑃)
19		simpllr 788	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥 ∈ 𝑃)
20		simplr 781	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑦 ∈ 𝑃)
21		cgrahlb.m	. . . 4 ⊢ − = (dist‘𝐺)
22		eqid 2763	. . . 4 ⊢ (cgrG‘𝐺) = (cgrG‘𝐺)
23		simpr 490	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
24	23	simpld 500	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)))
25	24	simprd 501	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐸 − 𝑥) = (𝐵 − 𝐴))
26	25	eqcomd 2769	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐵 − 𝐴) = (𝐸 − 𝑥))
27	1, 21, 2, 6, 10, 8, 16, 19, 26	tgcgrcomlr 28972	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴 − 𝐵) = (𝑥 − 𝐸))
28	23	simprd 501	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))
29	28	simprd 501	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐸 − 𝑦) = (𝐵 − 𝐶))
30	29	eqcomd 2769	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐵 − 𝐶) = (𝐸 − 𝑦))
31		eqid 2763	. . . . . 6 ⊢ (≤G‘𝐺) = (≤G‘𝐺)
32		cgrahlb.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴(𝐾‘𝐵)𝐶)
33	4, 32	syl 18	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐴(𝐾‘𝐵)𝐶)
34	1, 2, 3, 8, 12, 10, 6	ishlg 29098	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴(𝐾‘𝐵)𝐶 ↔ (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))))
35	33, 34	mpbid 235	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴 ≠ 𝐵 ∧ 𝐶 ≠ 𝐵 ∧ (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴))))
36	35	simp3d 1162	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴 ∈ (𝐵𝐼𝐶) ∨ 𝐶 ∈ (𝐵𝐼𝐴)))
37	24	simpld 500	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥(𝐾‘𝐸)𝐷)
38		cgrahlb.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷(𝐾‘𝐸)𝐹)
39	4, 38	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐷(𝐾‘𝐸)𝐹)
40	1, 2, 3, 19, 14, 18, 6, 16, 37, 39	hltr 29106	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥(𝐾‘𝐸)𝐹)
41	28	simpld 500	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑦(𝐾‘𝐸)𝐹)
42	1, 2, 3, 20, 18, 16, 6, 41	hlcomd 29100	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝐹(𝐾‘𝐸)𝑦)
43	1, 2, 3, 19, 18, 20, 6, 16, 40, 42	hltr 29106	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → 𝑥(𝐾‘𝐸)𝑦)
44	1, 2, 3, 19, 20, 16, 6	ishlg 29098	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥(𝐾‘𝐸)𝑦 ↔ (𝑥 ≠ 𝐸 ∧ 𝑦 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥)))))
45	43, 44	mpbid 235	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥 ≠ 𝐸 ∧ 𝑦 ≠ 𝐸 ∧ (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥))))
46	45	simp3d 1162	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝑥 ∈ (𝐸𝐼𝑦) ∨ 𝑦 ∈ (𝐸𝐼𝑥)))
47	1, 21, 2, 31, 6, 10, 8, 12, 16, 16, 19, 20, 36, 46, 26, 30	tgcgrsub2 29088	. . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐴 − 𝐶) = (𝑥 − 𝑦))
48	1, 21, 2, 6, 8, 12, 19, 20, 47	tgcgrcomlr 28972	. . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → (𝐶 − 𝐴) = (𝑦 − 𝑥))
49	1, 21, 22, 6, 8, 10, 12, 19, 16, 20, 27, 30, 48	trgcgr 29009	. . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrG‘𝐺)⟨“𝑥𝐸𝑦”⟩)
50	1, 2, 3, 6, 8, 10, 12, 14, 16, 18, 19, 20, 49, 37, 41	iscgrad 29348	. 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ 𝑦 ∈ 𝑃) ∧ ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))) → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)
51	1, 2, 3, 13, 17, 15, 5, 38	hlne1 29101	. . . . 5 ⊢ (𝜑 → 𝐷 ≠ 𝐸)
52	1, 2, 3, 7, 11, 9, 5, 32	hlne1 29101	. . . . . 6 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
53	52	necomd 3014	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
54	1, 2, 3, 15, 9, 7, 5, 13, 21, 51, 53	hlcgrex 29112	. . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)))
55	1, 2, 3, 13, 17, 15, 5, 38	hlne2 29102	. . . . 5 ⊢ (𝜑 → 𝐹 ≠ 𝐸)
56	1, 2, 3, 7, 11, 9, 5, 32	hlne2 29102	. . . . . 6 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
57	56	necomd 3014	. . . . 5 ⊢ (𝜑 → 𝐵 ≠ 𝐶)
58	1, 2, 3, 15, 9, 11, 5, 17, 21, 55, 57	hlcgrex 29112	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶)))
59	54, 58	jca 521	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
60		reeanv 3238	. . 3 ⊢ (∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))) ↔ (∃𝑥 ∈ 𝑃 (𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ ∃𝑦 ∈ 𝑃 (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
61	59, 60	sylibr 237	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 ∃𝑦 ∈ 𝑃 ((𝑥(𝐾‘𝐸)𝐷 ∧ (𝐸 − 𝑥) = (𝐵 − 𝐴)) ∧ (𝑦(𝐾‘𝐸)𝐹 ∧ (𝐸 − 𝑦) = (𝐵 − 𝐶))))
62	50, 61	r19.29vva 3226	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶”⟩(cgrA‘𝐺)⟨“𝐷𝐸𝐹”⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   class class class wbr 5106  ‘cfv 6558  (class class class)co 7443  ⟨“cs3 15021  Basecbs 17416  distcds 17466  TarskiGcstrkg 28919  Itvcitv 28925  cgrGccgrg 29003  ≤Gcleg 29075  hlGchlg 29093  cgrAccgra 29344

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-cnex 11283  ax-resscn 11284  ax-1cn 11285  ax-icn 11286  ax-addcl 11287  ax-addrcl 11288  ax-mulcl 11289  ax-mulrcl 11290  ax-mulcom 11291  ax-addass 11292  ax-mulass 11293  ax-distr 11294  ax-i2m1 11295  ax-1ne0 11296  ax-1rid 11297  ax-rnegex 11298  ax-rrecex 11299  ax-cnre 11300  ax-pre-lttri 11301  ax-pre-lttrn 11302  ax-pre-ltadd 11303  ax-pre-mulgt0 11304

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-oadd 8508  df-er 8745  df-map 8877  df-pm 8878  df-en 9002  df-dom 9003  df-sdom 9004  df-fin 9005  df-dju 10009  df-card 10047  df-pnf 11372  df-mnf 11373  df-xr 11374  df-ltxr 11375  df-le 11376  df-sub 11570  df-neg 11571  df-nn 12363  df-2 12432  df-3 12433  df-n0 12634  df-xnn0 12707  df-z 12721  df-uz 12993  df-fz 13667  df-fzo 13816  df-hash 14502  df-word 14686  df-concat 14743  df-s1 14770  df-s2 15027  df-s3 15028  df-trkgc 28940  df-trkgb 28941  df-trkgcb 28942  df-trkg 28945  df-cgrg 29004  df-leg 29076  df-hlg 29094  df-cgra 29345

This theorem is used by:  inagcgrasubhl  33079