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Theorem angmgmval 29424

Description: Explicit the value of the angle addition magma for a given geometry 𝐺. (Contributed by Thierry Arnoux, 31-Aug-2026.)

**Hypotheses**
Ref	Expression
angmgmval.p	⊢ 𝑃 = (Base‘𝐺)
angmgmval.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmgmval.i	⊢ 𝐼 = (Itv‘𝐺)
angmgmval.d	⊢ − = (dist‘𝐺)
angmgmval.c	⊢ ∼ = (cgrA‘𝐺)
angmgmval.l	⊢ 𝐿 = (LineG‘𝐺)
angmgmval.o	⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
angmgmval.j	⊢ 𝐽 = (AngMgm‘𝐺)
angmgmval.s	⊢ ≤ = (≤_∠‘𝐺)

**Assertion**
Ref	Expression
angmgmval	⊢ (𝐺 ∈ 𝑉 → 𝐽 = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ))

Distinct variable groups: 𝐴,𝑒,𝑓,𝑠 𝑒,𝐺,𝑓,𝑠 𝑃,𝑑,𝑒,𝑓,𝑠 Allowed substitution hints: 𝐴(𝑑) + (𝑒, 𝑓, 𝑠, 𝑑) ∼ (𝑒, 𝑓, 𝑠, 𝑑) 𝐺(𝑑) 𝐼(𝑒, 𝑓, 𝑠, 𝑑) 𝐽(𝑒, 𝑓, 𝑠, 𝑑) 𝐿(𝑒, 𝑓, 𝑠, 𝑑) ≤ (𝑒, 𝑓, 𝑠, 𝑑) − (𝑒, 𝑓, 𝑠, 𝑑) 𝑉(𝑒, 𝑓, 𝑠, 𝑑)

Proof of Theorem angmgmval Dummy variables 𝑎 𝑔 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		angmgmval.j	. 2 ⊢ 𝐽 = (AngMgm‘𝐺)
2		df-angmgm 29404	. . 3 ⊢ AngMgm = (𝑔 ∈ V ↦ ⦋(Base‘𝑔) / 𝑝⦌⦋{𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} / 𝑎⦌({⟨(Base‘ndx), 𝑎⟩, ⟨(+_g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤_∠‘𝑔)⟩} /_s (cgrA‘𝑔)))
3		fvexd 6923	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) ∈ V)
4		fveq2 6908	. . . . 5 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
5		angmgmval.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
6	4, 5	eqtr4di 2816	. . . 4 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝑃)
7		eqid 2763	. . . . . 6 ⊢ {𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} = {𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
8		ovexd 7478	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) → (𝑝 ↑_m (0..^3)) ∈ V)
9	7, 8	rabexd 5305	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) → {𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} ∈ V)
10		oveq1 7450	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑝 ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
11	10	adantl 487	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) → (𝑝 ↑_m (0..^3)) = (𝑃 ↑_m (0..^3)))
12	11	rabeqdv 3431	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) → {𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))})
13		angmgmval.a	. . . . . 6 ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
14	12, 13	eqtr4di 2816	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) → {𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} = 𝐴)
15		opeq2 4844	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ⟨(Base‘ndx), 𝑎⟩ = ⟨(Base‘ndx), 𝐴⟩)
16	15	adantl 487	. . . . . . 7 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ⟨(Base‘ndx), 𝑎⟩ = ⟨(Base‘ndx), 𝐴⟩)
17		simpr 490	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
18		fveq2 6908	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
19	18	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (LineG‘𝑔) = (LineG‘𝐺))
20		angmgmval.l	. . . . . . . . . . . . . 14 ⊢ 𝐿 = (LineG‘𝐺)
21	19, 20	eqtr4di 2816	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (LineG‘𝑔) = 𝐿)
22	21	oveqd 7460	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) = ((𝑒‘1)𝐿(𝑒‘2)))
23	22	eleq2d 2850	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ↔ (𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2))))
24		eqidd 2764	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑓‘0) = (𝑓‘0))
25		eqidd 2764	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑓‘1) = (𝑓‘1))
26		simplr 781	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → 𝑝 = 𝑃)
27		fveq2 6908	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝐺 → (cgrA‘𝑔) = (cgrA‘𝐺))
28		angmgmval.c	. . . . . . . . . . . . . . . . 17 ⊢ ∼ = (cgrA‘𝐺)
29	27, 28	eqtr4di 2816	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝐺 → (cgrA‘𝑔) = ∼ )
30	29	ad2antrr 739	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (cgrA‘𝑔) = ∼ )
31	30	breqd 5117	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ↔ ⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒))
32		fveq2 6908	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = (dist‘𝐺))
33		angmgmval.d	. . . . . . . . . . . . . . . . . 18 ⊢ − = (dist‘𝐺)
34	32, 33	eqtr4di 2816	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝐺 → (dist‘𝑔) = − )
35	34	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (dist‘𝑔) = − )
36	35	oveqd 7460	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑓‘1) − 𝑠))
37	35	oveqd 7460	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((𝑒‘1)(dist‘𝑔)(𝑒‘0)) = ((𝑒‘1) − (𝑒‘0)))
38	36, 37	eqeq12d 2779	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0)) ↔ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))
39	31, 38	anbi12d 644	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))) ↔ (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0)))))
40	26, 39	riotaeqbidv 7403	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0)))) = (℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0)))))
41	24, 25, 40	s3eqd 15043	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩ = ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩)
42		eqidd 2764	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑒‘0) = (𝑒‘0))
43		eqidd 2764	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑒‘1) = (𝑒‘1))
44	30	breqd 5117	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ↔ ⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓))
45	35	oveqd 7460	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑒‘1) − 𝑠))
46	35	oveqd 7460	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) = ((𝑓‘1) − (𝑓‘0)))
47	45, 46	eqeq12d 2779	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ↔ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0))))
48		fveq2 6908	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = (Itv‘𝐺))
49		angmgmval.i	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐼 = (Itv‘𝐺)
50	48, 49	eqtr4di 2816	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 = 𝐺 → (Itv‘𝑔) = 𝐼)
51	50	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (Itv‘𝑔) = 𝐼)
52	51	oveqd 7460	. . . . . . . . . . . . . . . 16 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑠(Itv‘𝑔)(𝑒‘0)) = (𝑠𝐼(𝑒‘0)))
53	22, 52	ineq12d 4121	. . . . . . . . . . . . . . 15 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) = (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))))
54	53	neeq1d 3018	. . . . . . . . . . . . . 14 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅ ↔ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))
55	44, 47, 54	3anbi123d 1465	. . . . . . . . . . . . 13 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ((⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅) ↔ (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅)))
56	26, 55	riotaeqbidv 7403	. . . . . . . . . . . 12 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅)) = (℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅)))
57	42, 43, 56	s3eqd 15043	. . . . . . . . . . 11 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩ = ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩)
58	23, 41, 57	ifbieq12d 4514	. . . . . . . . . 10 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩) = if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
59	17, 17, 58	mpoeq123dv 7518	. . . . . . . . 9 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩)) = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩)))
60		angmgmval.o	. . . . . . . . 9 ⊢ + = (𝑒 ∈ 𝐴, 𝑓 ∈ 𝐴 ↦ if((𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩ ∼ 𝑒 ∧ ((𝑓‘1) − 𝑠) = ((𝑒‘1) − (𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))”⟩))
61	59, 60	eqtr4di 2816	. . . . . . . 8 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩)) = + )
62	61	opeq2d 4850	. . . . . . 7 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ⟨(+_g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩ = ⟨(+_g‘ndx), + ⟩)
63		fveq2 6908	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (≤_∠‘𝑔) = (≤_∠‘𝐺))
64		angmgmval.s	. . . . . . . . . 10 ⊢ ≤ = (≤_∠‘𝐺)
65	63, 64	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (≤_∠‘𝑔) = ≤ )
66	65	opeq2d 4850	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → ⟨(le‘ndx), (≤_∠‘𝑔)⟩ = ⟨(le‘ndx), ≤ ⟩)
67	66	ad2antrr 739	. . . . . . 7 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ⟨(le‘ndx), (≤_∠‘𝑔)⟩ = ⟨(le‘ndx), ≤ ⟩)
68	16, 62, 67	tpeq123d 4709	. . . . . 6 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → {⟨(Base‘ndx), 𝑎⟩, ⟨(+_g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤_∠‘𝑔)⟩} = {⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩})
69	68, 30	oveq12d 7461	. . . . 5 ⊢ (((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) ∧ 𝑎 = 𝐴) → ({⟨(Base‘ndx), 𝑎⟩, ⟨(+_g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤_∠‘𝑔)⟩} /_s (cgrA‘𝑔)) = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ))
70	9, 14, 69	csbied2 3888	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑝 = 𝑃) → ⦋{𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} / 𝑎⦌({⟨(Base‘ndx), 𝑎⟩, ⟨(+_g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤_∠‘𝑔)⟩} /_s (cgrA‘𝑔)) = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ))
71	3, 6, 70	csbied2 3888	. . 3 ⊢ (𝑔 = 𝐺 → ⦋(Base‘𝑔) / 𝑝⦌⦋{𝑑 ∈ (𝑝 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))} / 𝑎⦌({⟨(Base‘ndx), 𝑎⟩, ⟨(+_g‘ndx), (𝑒 ∈ 𝑎, 𝑓 ∈ 𝑎 ↦ if((𝑒‘0) ∈ ((𝑒‘1)(LineG‘𝑔)(𝑒‘2)), ⟨“(𝑓‘0)(𝑓‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑓‘2)(𝑓‘1)𝑠”⟩(cgrA‘𝑔)𝑒 ∧ ((𝑓‘1)(dist‘𝑔)𝑠) = ((𝑒‘1)(dist‘𝑔)(𝑒‘0))))”⟩, ⟨“(𝑒‘0)(𝑒‘1)(℩𝑠 ∈ 𝑝 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩(cgrA‘𝑔)𝑓 ∧ ((𝑒‘1)(dist‘𝑔)𝑠) = ((𝑓‘1)(dist‘𝑔)(𝑓‘0)) ∧ (((𝑒‘1)(LineG‘𝑔)(𝑒‘2)) ∩ (𝑠(Itv‘𝑔)(𝑒‘0))) ≠ ∅))”⟩))⟩, ⟨(le‘ndx), (≤_∠‘𝑔)⟩} /_s (cgrA‘𝑔)) = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ))
72		elex 3475	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
73		ovexd 7478	. . 3 ⊢ (𝐺 ∈ 𝑉 → ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ) ∈ V)
74	2, 71, 72, 73	fvmptd3 7040	. 2 ⊢ (𝐺 ∈ 𝑉 → (AngMgm‘𝐺) = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ))
75	1, 74	eqtrid 2810	1 ⊢ (𝐺 ∈ 𝑉 → 𝐽 = ({⟨(Base‘ndx), 𝐴⟩, ⟨(+_g‘ndx), + ⟩, ⟨(le‘ndx), ≤ ⟩} /_s ∼ ))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 {crab 3416 Vcvv 3454 ⦋csb 3851 ∩ cin 4092 ∅c0 4281 ifcif 4485 {ctp 4592 ⟨cop 4838 class class class wbr 5106 ‘cfv 6558 ℩crio 7399 (class class class)co 7443 ∈ cmpo 7445 ↑_m cmap 8875 0cc0 11227 1c1 11228 2c2 12424 3c3 12425 ..^cfzo 13815 ⟨“cs3 15021 ndxcnx 17400 Basecbs 17416 +_gcplusg 17457 lecple 17464 distcds 17466 /_s cqus 17706 Itvcitv 28925 LineGclng 28926 cgrAccgra 29344 ≤_∠cleag 29385 AngMgmcangmgm 29403

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-sep 5252 ax-nul 5263 ax-pr 5395

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 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-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 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-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6512 df-fun 6560 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-s1 14770 df-s2 15027 df-s3 15028 df-angmgm 29404

This theorem is used by: angmgmlem 29425 angmgmbas 29428

W3C validator