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Theorem angmndaddov1 29259

Description: Value of the addition operation in the angle addition monoid, in case the first angle is neither zero nor flat. (Contributed by Thierry Arnoux, 23-Aug-2026.)

**Hypotheses**
Ref	Expression
angmndadd.p	⊢ 𝑃 = (Base‘𝐺)
angmndadd.a	⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
angmndadd.i	⊢ 𝐼 = (Itv‘𝐺)
angmndadd.d	⊢ − = (dist‘𝐺)
angmndadd.c	⊢ ∼ = (cgrA‘𝐺)
angmndadd.l	⊢ 𝐿 = (LineG‘𝐺)
angmndadd.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
angmndaddov.u	⊢ (𝜑 → 𝑈 ∈ 𝑃)
angmndaddov.v	⊢ (𝜑 → 𝑉 ∈ 𝑃)
angmndaddov.w	⊢ (𝜑 → 𝑊 ∈ 𝑃)
angmndaddov.x	⊢ (𝜑 → 𝑋 ∈ 𝑃)
angmndaddov.y	⊢ (𝜑 → 𝑌 ∈ 𝑃)
angmndaddov.z	⊢ (𝜑 → 𝑍 ∈ 𝑃)
angmndaddeu.1	⊢ (𝜑 → 𝑈 ≠ 𝑉)
angmndaddeu.2	⊢ (𝜑 → 𝑉 ≠ 𝑊)
angmndaddeu.3	⊢ (𝜑 → 𝑋 ≠ 𝑌)
angmndaddeu.4	⊢ (𝜑 → 𝑌 ≠ 𝑍)
angmndaddov.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))) ≠ ∅))”⟩))
angmndaddov1.x	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
angmndaddov1.s	⊢ (𝜑 → 𝑆 ∈ 𝑃)
angmndaddov1.1	⊢ (𝜑 → ⟨“𝑍𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
angmndaddov1.2	⊢ (𝜑 → (𝑌 − 𝑆) = (𝑉 − 𝑈))
angmndaddov1.3	⊢ (𝜑 → ((𝑌𝐿𝑍) ∩ (𝑆𝐼𝑋)) ≠ ∅)

**Assertion**
Ref	Expression
angmndaddov1	⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩ + ⟨“𝑈𝑉𝑊”⟩) = ⟨“𝑋𝑌𝑆”⟩)

Distinct variable groups: − ,𝑑,𝑠 ∼ ,𝑠 𝐺,𝑑,𝑠 𝐼,𝑑,𝑠 𝐿,𝑑,𝑒,𝑓,𝑠 𝑃,𝑑,𝑠 𝑆,𝑑,𝑒,𝑓,𝑠 𝑈,𝑑,𝑒,𝑓,𝑠 𝑉,𝑑,𝑒,𝑓,𝑠 𝑊,𝑑,𝑒,𝑓,𝑠 𝑋,𝑑,𝑒,𝑓,𝑠 𝑌,𝑑,𝑒,𝑓,𝑠 𝑍,𝑑,𝑒,𝑓,𝑠 𝜑,𝑑,𝑒,𝑓,𝑠 Allowed substitution hints: 𝐴(𝑒, 𝑓, 𝑠, 𝑑) 𝑃(𝑒, 𝑓) + (𝑒, 𝑓, 𝑠, 𝑑) ∼ (𝑒, 𝑓, 𝑑) 𝐺(𝑒, 𝑓) 𝐼(𝑒, 𝑓) − (𝑒, 𝑓)

**Proof of Theorem angmndaddov1**
Step	Hyp	Ref	Expression
1		angmndaddov.o	. . 3 ⊢ + = (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))) ≠ ∅))”⟩))
2	1	a1i 11	. 2 ⊢ (𝜑 → + = (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))) ≠ ∅))”⟩)))
3		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑒 = ⟨“𝑋𝑌𝑍”⟩)
4	3	fveq1d 6884	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘0) = (⟨“𝑋𝑌𝑍”⟩‘0))
5		angmndaddov.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑃)
6		s3fv0 14962	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝑃 → (⟨“𝑋𝑌𝑍”⟩‘0) = 𝑋)
7	5, 6	syl 18	. . . . . . . 8 ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩‘0) = 𝑋)
8	7	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (⟨“𝑋𝑌𝑍”⟩‘0) = 𝑋)
9	4, 8	eqtrd 2797	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘0) = 𝑋)
10		angmndaddov1.x	. . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
11	10	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ¬ 𝑋 ∈ (𝑌𝐿𝑍))
12	3	fveq1d 6884	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘1) = (⟨“𝑋𝑌𝑍”⟩‘1))
13		angmndaddov.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ 𝑃)
14		s3fv1 14963	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝑃 → (⟨“𝑋𝑌𝑍”⟩‘1) = 𝑌)
15	13, 14	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩‘1) = 𝑌)
16	15	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (⟨“𝑋𝑌𝑍”⟩‘1) = 𝑌)
17	12, 16	eqtrd 2797	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘1) = 𝑌)
18	3	fveq1d 6884	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘2) = (⟨“𝑋𝑌𝑍”⟩‘2))
19		angmndaddov.z	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑍 ∈ 𝑃)
20		s3fv2 14964	. . . . . . . . . . 11 ⊢ (𝑍 ∈ 𝑃 → (⟨“𝑋𝑌𝑍”⟩‘2) = 𝑍)
21	19, 20	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩‘2) = 𝑍)
22	21	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (⟨“𝑋𝑌𝑍”⟩‘2) = 𝑍)
23	18, 22	eqtrd 2797	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘2) = 𝑍)
24	17, 23	oveq12d 7434	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((𝑒‘1)𝐿(𝑒‘2)) = (𝑌𝐿𝑍))
25	11, 24	neleqtrrd 2885	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ¬ 𝑋 ∈ ((𝑒‘1)𝐿(𝑒‘2)))
26	9, 25	eqneltrd 2882	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ¬ (𝑒‘0) ∈ ((𝑒‘1)𝐿(𝑒‘2)))
27	26	iffalsed 4496	. . . 4 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 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))) ≠ ∅))”⟩) = ⟨“(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))) ≠ ∅))”⟩)
28		angmndaddov1.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ 𝑃)
29	28	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑆 ∈ 𝑃)
30		angmndadd.p	. . . . . . . 8 ⊢ 𝑃 = (Base‘𝐺)
31		angmndadd.a	. . . . . . . 8 ⊢ 𝐴 = {𝑑 ∈ (𝑃 ↑_m (0..^3)) ∣ ((𝑑‘0) ≠ (𝑑‘1) ∧ (𝑑‘1) ≠ (𝑑‘2))}
32		angmndadd.i	. . . . . . . 8 ⊢ 𝐼 = (Itv‘𝐺)
33		angmndadd.d	. . . . . . . 8 ⊢ − = (dist‘𝐺)
34		angmndadd.c	. . . . . . . 8 ⊢ ∼ = (cgrA‘𝐺)
35		angmndadd.l	. . . . . . . 8 ⊢ 𝐿 = (LineG‘𝐺)
36		angmndadd.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
37	36	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝐺 ∈ TarskiG)
38		angmndaddov.u	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝑃)
39	38	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑈 ∈ 𝑃)
40		angmndaddov.v	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ 𝑃)
41	40	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑉 ∈ 𝑃)
42		angmndaddov.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ 𝑃)
43	42	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑊 ∈ 𝑃)
44	5	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑋 ∈ 𝑃)
45	13	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑌 ∈ 𝑃)
46	17, 45	eqeltrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘1) ∈ 𝑃)
47	19	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑍 ∈ 𝑃)
48	23, 47	eqeltrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘2) ∈ 𝑃)
49		angmndaddeu.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
50	49	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑈 ≠ 𝑉)
51		angmndaddeu.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ≠ 𝑊)
52	51	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑉 ≠ 𝑊)
53		angmndaddeu.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ≠ 𝑌)
54	53	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑋 ≠ 𝑌)
55	54, 17	neeqtrrd 3031	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑋 ≠ (𝑒‘1))
56		angmndaddeu.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ≠ 𝑍)
57	56	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑌 ≠ 𝑍)
58	17, 57	eqnetrd 3024	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘1) ≠ 𝑍)
59	58, 23	neeqtrrd 3031	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑒‘1) ≠ (𝑒‘2))
60	30, 31, 32, 33, 34, 35, 37, 39, 41, 43, 44, 46, 48, 50, 52, 55, 59, 25	angmndaddov1lem 29257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ∃!𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ ((𝑒‘1) − 𝑠) = (𝑉 − 𝑈) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼𝑋)) ≠ ∅))
61		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑓 = ⟨“𝑈𝑉𝑊”⟩)
62	61	breq2d 5119	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ↔ ⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩))
63	61	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑓‘1) = (⟨“𝑈𝑉𝑊”⟩‘1))
64		s3fv1 14963	. . . . . . . . . . . . . 14 ⊢ (𝑉 ∈ 𝑃 → (⟨“𝑈𝑉𝑊”⟩‘1) = 𝑉)
65	40, 64	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑈𝑉𝑊”⟩‘1) = 𝑉)
66	65	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (⟨“𝑈𝑉𝑊”⟩‘1) = 𝑉)
67	63, 66	eqtrd 2797	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑓‘1) = 𝑉)
68	61	fveq1d 6884	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑓‘0) = (⟨“𝑈𝑉𝑊”⟩‘0))
69		s3fv0 14962	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ 𝑃 → (⟨“𝑈𝑉𝑊”⟩‘0) = 𝑈)
70	38, 69	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (⟨“𝑈𝑉𝑊”⟩‘0) = 𝑈)
71	70	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (⟨“𝑈𝑉𝑊”⟩‘0) = 𝑈)
72	68, 71	eqtrd 2797	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑓‘0) = 𝑈)
73	67, 72	oveq12d 7434	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((𝑓‘1) − (𝑓‘0)) = (𝑉 − 𝑈))
74	73	eqeq2d 2773	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ↔ ((𝑒‘1) − 𝑠) = (𝑉 − 𝑈)))
75	9	oveq2d 7432	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑠𝐼(𝑒‘0)) = (𝑠𝐼𝑋))
76	75	ineq2d 4169	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) = (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼𝑋)))
77	76	neeq1d 3016	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅ ↔ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼𝑋)) ≠ ∅))
78	62, 74, 77	3anbi123d 1464	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅) ↔ (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ ((𝑒‘1) − 𝑠) = (𝑉 − 𝑈) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼𝑋)) ≠ ∅)))
79	78	reubidv 3383	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (∃!𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅) ↔ ∃!𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩ ∧ ((𝑒‘1) − 𝑠) = (𝑉 − 𝑈) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼𝑋)) ≠ ∅)))
80	60, 79	mpbird 260	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ∃!𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅))
81		angmndaddov1.1	. . . . . . . 8 ⊢ (𝜑 → ⟨“𝑍𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
82	81	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ⟨“𝑍𝑌𝑆”⟩ ∼ ⟨“𝑈𝑉𝑊”⟩)
83		eqidd 2763	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 𝑆 = 𝑆)
84	23, 17, 83	s3eqd 14935	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩ = ⟨“𝑍𝑌𝑆”⟩)
85	82, 84, 61	3brtr4d 5141	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩ ∼ 𝑓)
86		angmndaddov1.2	. . . . . . . 8 ⊢ (𝜑 → (𝑌 − 𝑆) = (𝑉 − 𝑈))
87	86	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑌 − 𝑆) = (𝑉 − 𝑈))
88	17	oveq1d 7431	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((𝑒‘1) − 𝑆) = (𝑌 − 𝑆))
89	87, 88, 73	3eqtr4d 2807	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((𝑒‘1) − 𝑆) = ((𝑓‘1) − (𝑓‘0)))
90	9	oveq2d 7432	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (𝑆𝐼(𝑒‘0)) = (𝑆𝐼𝑋))
91	24, 90	ineq12d 4170	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(𝑒‘0))) = ((𝑌𝐿𝑍) ∩ (𝑆𝐼𝑋)))
92		angmndaddov1.3	. . . . . . . 8 ⊢ (𝜑 → ((𝑌𝐿𝑍) ∩ (𝑆𝐼𝑋)) ≠ ∅)
93	92	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ((𝑌𝐿𝑍) ∩ (𝑆𝐼𝑋)) ≠ ∅)
94	91, 93	eqnetrd 3024	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(𝑒‘0))) ≠ ∅)
95		eqidd 2763	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑒‘2) = (𝑒‘2))
96		eqidd 2763	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑒‘1) = (𝑒‘1))
97		id 23	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
98	95, 96, 97	s3eqd 14935	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ = ⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩)
99	98	breq1d 5117	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ↔ ⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩ ∼ 𝑓))
100		oveq2 7424	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → ((𝑒‘1) − 𝑠) = ((𝑒‘1) − 𝑆))
101	100	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ↔ ((𝑒‘1) − 𝑆) = ((𝑓‘1) − (𝑓‘0))))
102		oveq1 7423	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑠𝐼(𝑒‘0)) = (𝑆𝐼(𝑒‘0)))
103	102	ineq2d 4169	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) = (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(𝑒‘0))))
104	103	neeq1d 3016	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → ((((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅ ↔ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(𝑒‘0))) ≠ ∅))
105	99, 101, 104	3anbi123d 1464	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → ((⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅) ↔ (⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑆) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(𝑒‘0))) ≠ ∅)))
106	105	riota2 7398	. . . . . . 7 ⊢ ((𝑆 ∈ 𝑃 ∧ ∃!𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅)) → ((⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑆) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(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))) ≠ ∅)) = 𝑆))
107	106	biimpa 482	. . . . . 6 ⊢ (((𝑆 ∈ 𝑃 ∧ ∃!𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅)) ∧ (⟨“(𝑒‘2)(𝑒‘1)𝑆”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑆) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑆𝐼(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))) ≠ ∅)) = 𝑆)
108	29, 80, 85, 89, 94, 107	syl23anc 1404	. . . . 5 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → (℩𝑠 ∈ 𝑃 (⟨“(𝑒‘2)(𝑒‘1)𝑠”⟩ ∼ 𝑓 ∧ ((𝑒‘1) − 𝑠) = ((𝑓‘1) − (𝑓‘0)) ∧ (((𝑒‘1)𝐿(𝑒‘2)) ∩ (𝑠𝐼(𝑒‘0))) ≠ ∅)) = 𝑆)
109	9, 17, 108	s3eqd 14935	. . . 4 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → ⟨“(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))) ≠ ∅))”⟩ = ⟨“𝑋𝑌𝑆”⟩)
110	27, 109	eqtrd 2797	. . 3 ⊢ (((𝜑 ∧ 𝑒 = ⟨“𝑋𝑌𝑍”⟩) ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩) → 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))) ≠ ∅))”⟩) = ⟨“𝑋𝑌𝑆”⟩)
111	110	anasss 472	. 2 ⊢ ((𝜑 ∧ (𝑒 = ⟨“𝑋𝑌𝑍”⟩ ∧ 𝑓 = ⟨“𝑈𝑉𝑊”⟩)) → 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))) ≠ ∅))”⟩) = ⟨“𝑋𝑌𝑆”⟩)
112	30	fvexi 6896	. . . 4 ⊢ 𝑃 ∈ V
113	112	a1i 11	. . 3 ⊢ (𝜑 → 𝑃 ∈ V)
114	31, 113, 5, 13, 19, 53, 56	elcgrabasrd 29249	. 2 ⊢ (𝜑 → ⟨“𝑋𝑌𝑍”⟩ ∈ 𝐴)
115	31, 113, 38, 40, 42, 49, 51	elcgrabasrd 29249	. 2 ⊢ (𝜑 → ⟨“𝑈𝑉𝑊”⟩ ∈ 𝐴)
116	86	eqcomd 2768	. . . 4 ⊢ (𝜑 → (𝑉 − 𝑈) = (𝑌 − 𝑆))
117	49	necomd 3012	. . . 4 ⊢ (𝜑 → 𝑉 ≠ 𝑈)
118	30, 33, 32, 36, 40, 38, 13, 28, 116, 117	tgcgrneq 28820	. . 3 ⊢ (𝜑 → 𝑌 ≠ 𝑆)
119	31, 113, 5, 13, 28, 53, 118	elcgrabasrd 29249	. 2 ⊢ (𝜑 → ⟨“𝑋𝑌𝑆”⟩ ∈ 𝐴)
120	2, 111, 114, 115, 119	ovmpod 7568	1 ⊢ (𝜑 → (⟨“𝑋𝑌𝑍”⟩ + ⟨“𝑈𝑉𝑊”⟩) = ⟨“𝑋𝑌𝑆”⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∃!wreu 3365 {crab 3414 Vcvv 3453 ∩ cin 3901 ∅c0 4282 ifcif 4485 class class class wbr 5107 ‘cfv 6537 ℩crio 7372 (class class class)co 7416 ∈ cmpo 7418 ↑_m cmap 8829 0cc0 11125 1c1 11126 2c2 12320 3c3 12321 ..^cfzo 13709 ⟨“cs3 14913 Basecbs 17303 distcds 17353 TarskiGcstrkg 28764 Itvcitv 28770 LineGclng 28771 cgrAccgra 29189

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 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-oadd 8462 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-dju 9909 df-card 9947 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-n0 12530 df-xnn0 12603 df-z 12617 df-uz 12889 df-fz 13562 df-fzo 13710 df-hash 14395 df-word 14579 df-concat 14636 df-s1 14663 df-s2 14919 df-s3 14920 df-trkgc 28785 df-trkgb 28786 df-trkgcb 28787 df-trkgld 28789 df-trkg 28790 df-cgrg 28849 df-ismt 28871 df-leg 28921 df-hlg 28939 df-mir 29000 df-rag 29044 df-perpg 29046 df-hpg 29111 df-mid 29154 df-lmi 29155 df-cgra 29190

This theorem is used by: angmndaddcpbl 29261

W3C validator