signstres - Mathbox for Thierry Arnoux

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Theorem signstres 29978

Description: Restriction of a zero skipping sign to a subword. (Contributed by Thierry Arnoux, 11-Oct-2018.)

**Hypotheses**
Ref	Expression
signsv.p	⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
signsv.w	⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
signsv.t	⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
signsv.v	⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))

**Assertion**
Ref	Expression
signstres	⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) = (𝑇‘(𝐹 ↾ (0..^𝑁))))

Distinct variable groups: 𝑎,𝑏, ⨣ 𝑓,𝑖,𝑛,𝐹 𝑓,𝑊,𝑖,𝑛 𝑓,𝑁,𝑖,𝑛 Allowed substitution hints: ⨣ (𝑓,𝑖,𝑗,𝑛) 𝑇(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏) 𝐹(𝑗,𝑎,𝑏) 𝑁(𝑗,𝑎,𝑏) 𝑉(𝑓,𝑖,𝑗,𝑛,𝑎,𝑏) 𝑊(𝑗,𝑎,𝑏)

Proof of Theorem signstres Dummy variables 𝑔 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		signsv.p	. . . . . . . 8 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
2		signsv.w	. . . . . . . 8 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
3		signsv.t	. . . . . . . 8 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(#‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
4		signsv.v	. . . . . . . 8 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(#‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
5	1, 2, 3, 4	signstf 29969	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ)
6		wrdf 13165	. . . . . . 7 ⊢ ((𝑇‘𝐹) ∈ Word ℝ → (𝑇‘𝐹):(0..^(#‘(𝑇‘𝐹)))⟶ℝ)
7		ffn 5958	. . . . . . 7 ⊢ ((𝑇‘𝐹):(0..^(#‘(𝑇‘𝐹)))⟶ℝ → (𝑇‘𝐹) Fn (0..^(#‘(𝑇‘𝐹))))
8	5, 6, 7	3syl 18	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) Fn (0..^(#‘(𝑇‘𝐹))))
9	1, 2, 3, 4	signstlen 29970	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (#‘(𝑇‘𝐹)) = (#‘𝐹))
10	9	oveq2d 6565	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(#‘(𝑇‘𝐹))) = (0..^(#‘𝐹)))
11	10	fneq2d 5896	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘𝐹) Fn (0..^(#‘(𝑇‘𝐹))) ↔ (𝑇‘𝐹) Fn (0..^(#‘𝐹))))
12	8, 11	mpbid 221	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) Fn (0..^(#‘𝐹)))
13		fnresin 28812	. . . . 5 ⊢ ((𝑇‘𝐹) Fn (0..^(#‘𝐹)) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(#‘𝐹)) ∩ (0..^𝑁)))
14	12, 13	syl 17	. . . 4 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(#‘𝐹)) ∩ (0..^𝑁)))
15	14	adantr 480	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(#‘𝐹)) ∩ (0..^𝑁)))
16		elfzuz3 12210	. . . . . 6 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → (#‘𝐹) ∈ (ℤ_≥‘𝑁))
17		fzoss2 12365	. . . . . 6 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
18	16, 17	syl 17	. . . . 5 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
19	18	adantl 481	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
20		incom 3767	. . . . . 6 ⊢ ((0..^𝑁) ∩ (0..^(#‘𝐹))) = ((0..^(#‘𝐹)) ∩ (0..^𝑁))
21		df-ss 3554	. . . . . . 7 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) ↔ ((0..^𝑁) ∩ (0..^(#‘𝐹))) = (0..^𝑁))
22	21	biimpi 205	. . . . . 6 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) → ((0..^𝑁) ∩ (0..^(#‘𝐹))) = (0..^𝑁))
23	20, 22	syl5eqr 2658	. . . . 5 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) → ((0..^(#‘𝐹)) ∩ (0..^𝑁)) = (0..^𝑁))
24	23	fneq2d 5896	. . . 4 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) → (((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(#‘𝐹)) ∩ (0..^𝑁)) ↔ ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁)))
25	19, 24	syl 17	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(#‘𝐹)) ∩ (0..^𝑁)) ↔ ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁)))
26	15, 25	mpbid 221	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁))
27		wrdres 29943	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ)
28	1, 2, 3, 4	signstf 29969	. . . 4 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ)
29		wrdf 13165	. . . 4 ⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ)
30		ffn 5958	. . . 4 ⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁))))))
31	27, 28, 29, 30	4syl 19	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁))))))
32	1, 2, 3, 4	signstlen 29970	. . . . . . 7 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = (#‘(𝐹 ↾ (0..^𝑁))))
33	27, 32	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = (#‘(𝐹 ↾ (0..^𝑁))))
34		wrdfn 13174	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 𝐹 Fn (0..^(#‘𝐹)))
35		fnssres 5918	. . . . . . . 8 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
36	34, 18, 35	syl2an 493	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
37		hashfn 13025	. . . . . . 7 ⊢ ((𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁) → (#‘(𝐹 ↾ (0..^𝑁))) = (#‘(0..^𝑁)))
38	36, 37	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝐹 ↾ (0..^𝑁))) = (#‘(0..^𝑁)))
39		elfznn0 12302	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → 𝑁 ∈ ℕ₀)
40		hashfzo0 13077	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (#‘(0..^𝑁)) = 𝑁)
41	39, 40	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → (#‘(0..^𝑁)) = 𝑁)
42	41	adantl 481	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(0..^𝑁)) = 𝑁)
43	33, 38, 42	3eqtrd 2648	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = 𝑁)
44	43	oveq2d 6565	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) = (0..^𝑁))
45	44	fneq2d 5896	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) ↔ (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁)))
46	31, 45	mpbid 221	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁))
47		fvres 6117	. . . . 5 ⊢ (𝑚 ∈ (0..^𝑁) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚))
48	47	ad3antlr 763	. . . 4 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚))
49		simpr 476	. . . . . 6 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
50	49	fveq2d 6107	. . . . 5 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (𝑇‘𝐹) = (𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔)))
51	50	fveq1d 6105	. . . 4 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → ((𝑇‘𝐹)‘𝑚) = ((𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))‘𝑚))
52	27	ad3antrrr 762	. . . . 5 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ)
53		simplr 788	. . . . 5 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → 𝑔 ∈ Word ℝ)
54	38, 42	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
55	54	oveq2d 6565	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (0..^(#‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁))
56	55	eleq2d 2673	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝑚 ∈ (0..^(#‘(𝐹 ↾ (0..^𝑁)))) ↔ 𝑚 ∈ (0..^𝑁)))
57	56	biimpar 501	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → 𝑚 ∈ (0..^(#‘(𝐹 ↾ (0..^𝑁)))))
58	57	ad2antrr 758	. . . . 5 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → 𝑚 ∈ (0..^(#‘(𝐹 ↾ (0..^𝑁)))))
59	1, 2, 3, 4	signstfvc 29977	. . . . 5 ⊢ (((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ ∧ 𝑔 ∈ Word ℝ ∧ 𝑚 ∈ (0..^(#‘(𝐹 ↾ (0..^𝑁))))) → ((𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚))
60	52, 53, 58, 59	syl3anc 1318	. . . 4 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → ((𝑇‘((𝐹 ↾ (0..^𝑁)) ++ 𝑔))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚))
61	48, 51, 60	3eqtrd 2648	. . 3 ⊢ (((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) ∧ 𝑔 ∈ Word ℝ) ∧ 𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔)) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚))
62		wrdsplex 29944	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
63	62	adantr 480	. . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
64	61, 63	r19.29a 3060	. 2 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘(𝐹 ↾ (0..^𝑁)))‘𝑚))
65	26, 46, 64	eqfnfvd 6222	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) = (𝑇‘(𝐹 ↾ (0..^𝑁))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 ifcif 4036 {cpr 4127 {ctp 4129 ⟨cop 4131 ↦ cmpt 4643 ↾ cres 5040 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 0cc0 9815 1c1 9816 − cmin 10145 -cneg 10146 ℕ₀cn0 11169 ℤ_≥cuz 11563 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 ++ cconcat 13148 sgncsgn 13674 Σcsu 14264 ndxcnx 15692 Basecbs 15695 +_gcplusg 15768 Σ_g cgsu 15924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-xnn0 11241 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-lsw 13155 df-concat 13156 df-s1 13157 df-substr 13158 df-sgn 13675 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118

This theorem is referenced by: (None)

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