signstres - Mathbox for Thierry Arnoux

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Theorem signstres 30652

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 30643	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ)
6		wrdf 13310	. . . . . . 7 ⊢ ((𝑇‘𝐹) ∈ Word ℝ → (𝑇‘𝐹):(0..^(#‘(𝑇‘𝐹)))⟶ℝ)
7		ffn 6045	. . . . . . 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 30644	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (#‘(𝑇‘𝐹)) = (#‘𝐹))
10	9	oveq2d 6666	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(#‘(𝑇‘𝐹))) = (0..^(#‘𝐹)))
11	10	fneq2d 5982	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘𝐹) Fn (0..^(#‘(𝑇‘𝐹))) ↔ (𝑇‘𝐹) Fn (0..^(#‘𝐹))))
12	8, 11	mpbid 222	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) Fn (0..^(#‘𝐹)))
13		fnresin 29430	. . . . 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 481	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(#‘𝐹)) ∩ (0..^𝑁)))
16		elfzuz3 12339	. . . . . 6 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → (#‘𝐹) ∈ (ℤ_≥‘𝑁))
17		fzoss2 12496	. . . . . 6 ⊢ ((#‘𝐹) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
18	16, 17	syl 17	. . . . 5 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
19	18	adantl 482	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
20		incom 3805	. . . . . 6 ⊢ ((0..^𝑁) ∩ (0..^(#‘𝐹))) = ((0..^(#‘𝐹)) ∩ (0..^𝑁))
21		df-ss 3588	. . . . . . 7 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) ↔ ((0..^𝑁) ∩ (0..^(#‘𝐹))) = (0..^𝑁))
22	21	biimpi 206	. . . . . 6 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) → ((0..^𝑁) ∩ (0..^(#‘𝐹))) = (0..^𝑁))
23	20, 22	syl5eqr 2670	. . . . 5 ⊢ ((0..^𝑁) ⊆ (0..^(#‘𝐹)) → ((0..^(#‘𝐹)) ∩ (0..^𝑁)) = (0..^𝑁))
24	23	fneq2d 5982	. . . 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 222	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁))
27		wrdres 30617	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ)
28	1, 2, 3, 4	signstf 30643	. . . 4 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ)
29		wrdf 13310	. . . 4 ⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ)
30		ffn 6045	. . . 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 30644	. . . . . . 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 13319	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 𝐹 Fn (0..^(#‘𝐹)))
35		fnssres 6004	. . . . . . . 8 ⊢ ((𝐹 Fn (0..^(#‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
36	34, 18, 35	syl2an 494	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
37		hashfn 13164	. . . . . . 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 12433	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → 𝑁 ∈ ℕ₀)
40		hashfzo0 13217	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (#‘(0..^𝑁)) = 𝑁)
41	39, 40	syl 17	. . . . . . 7 ⊢ (𝑁 ∈ (0...(#‘𝐹)) → (#‘(0..^𝑁)) = 𝑁)
42	41	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(0..^𝑁)) = 𝑁)
43	33, 38, 42	3eqtrd 2660	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = 𝑁)
44	43	oveq2d 6666	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (0..^(#‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) = (0..^𝑁))
45	44	fneq2d 5982	. . 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 222	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁))
47		fvres 6207	. . . . 5 ⊢ (𝑚 ∈ (0..^𝑁) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚))
48	47	ad3antlr 767	. . . 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 477	. . . . . 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 6195	. . . . 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 6193	. . . 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 766	. . . . 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 792	. . . . 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 2656	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
55	54	oveq2d 6666	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (0..^(#‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁))
56	55	eleq2d 2687	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → (𝑚 ∈ (0..^(#‘(𝐹 ↾ (0..^𝑁)))) ↔ 𝑚 ∈ (0..^𝑁)))
57	56	biimpar 502	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → 𝑚 ∈ (0..^(#‘(𝐹 ↾ (0..^𝑁)))))
58	57	ad2antrr 762	. . . . 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 30651	. . . . 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 1326	. . . 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 2660	. . 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 30618	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
63	62	adantr 481	. . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(#‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
64	61, 63	r19.29a 3078	. 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 6314	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 ifcif 4086 {cpr 4179 {ctp 4181 ⟨cop 4183 ↦ cmpt 4729 ↾ cres 5116 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℝcr 9935 0cc0 9936 1c1 9937 − cmin 10266 -cneg 10267 ℕ₀cn0 11292 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 ++ cconcat 13293 sgncsgn 13826 Σcsu 14416 ndxcnx 15854 Basecbs 15857 +_gcplusg 15941 Σ_g cgsu 16101

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-sgn 13827 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-gsum 16103 df-mgm 17242 df-sgrp 17284 df-mnd 17295

This theorem is referenced by: (None)

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