signstres - Mathbox for Thierry Arnoux

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Theorem signstres 34539

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 34530	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ)
6		wrdf 14459	. . . . . . 7 ⊢ ((𝑇‘𝐹) ∈ Word ℝ → (𝑇‘𝐹):(0..^(♯‘(𝑇‘𝐹)))⟶ℝ)
7		ffn 6670	. . . . . . 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 34531	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(𝑇‘𝐹)) = (♯‘𝐹))
10	9	oveq2d 7385	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(𝑇‘𝐹))) = (0..^(♯‘𝐹)))
11	10	fneq2d 6594	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹))) ↔ (𝑇‘𝐹) Fn (0..^(♯‘𝐹))))
12	8, 11	mpbid 232	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) Fn (0..^(♯‘𝐹)))
13		fnresin 32523	. . . . 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 13458	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ_≥‘𝑁))
17		fzoss2 13624	. . . . . 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 4168	. . . . . 6 ⊢ ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = ((0..^(♯‘𝐹)) ∩ (0..^𝑁))
21		dfss2 3929	. . . . . . 7 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) ↔ ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = (0..^𝑁))
22	21	biimpi 216	. . . . . 6 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) → ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = (0..^𝑁))
23	20, 22	eqtr3id 2778	. . . . 5 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) → ((0..^(♯‘𝐹)) ∩ (0..^𝑁)) = (0..^𝑁))
24	23	fneq2d 6594	. . . 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 232	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁))
27		wrdres 32829	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ)
28	1, 2, 3, 4	signstf 34530	. . . 4 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ)
29		wrdf 14459	. . . 4 ⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ)
30		ffn 6670	. . . 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 34531	. . . . . . 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 14469	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 𝐹 Fn (0..^(♯‘𝐹)))
35		fnssres 6623	. . . . . . . 8 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
36	34, 18, 35	syl2an 596	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
37		hashfn 14316	. . . . . . 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 13557	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → 𝑁 ∈ ℕ₀)
40		hashfzo0 14371	. . . . . . . 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 2768	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = 𝑁)
44	43	oveq2d 7385	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) = (0..^𝑁))
45	44	fneq2d 6594	. . 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 232	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁))
47		fvres 6859	. . . . 5 ⊢ (𝑚 ∈ (0..^𝑁) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚))
48	47	ad3antlr 731	. . . 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 484	. . . . . 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 6844	. . . . 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 6842	. . . 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 730	. . . . 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 768	. . . . 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 2764	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
55	54	oveq2d 7385	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁))
56	55	eleq2d 2814	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝑚 ∈ (0..^(♯‘(𝐹 ↾ (0..^𝑁)))) ↔ 𝑚 ∈ (0..^𝑁)))
57	56	biimpar 477	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → 𝑚 ∈ (0..^(♯‘(𝐹 ↾ (0..^𝑁)))))
58	57	ad2antrr 726	. . . . 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 34538	. . . . 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 1373	. . . 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 2768	. . 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 32830	. . . 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 3141	. 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 6988	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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 ∩ cin 3910 ⊆ wss 3911 ifcif 4484 {cpr 4587 {ctp 4589 ⟨cop 4591 ↦ cmpt 5183 ↾ cres 5633 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 ℝcr 11043 0cc0 11044 1c1 11045 − cmin 11381 -cneg 11382 ℕ₀cn0 12418 ℤ_≥cuz 12769 ...cfz 13444 ..^cfzo 13591 ♯chash 14271 Word cword 14454 ++ cconcat 14511 sgncsgn 15028 Σcsu 15628 ndxcnx 17139 Basecbs 17155 +_gcplusg 17196 Σ_g cgsu 17379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-n0 12419 df-xnn0 12492 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-seq 13943 df-hash 14272 df-word 14455 df-lsw 14504 df-concat 14512 df-s1 14537 df-substr 14582 df-pfx 14612 df-sgn 15029 df-struct 17093 df-slot 17128 df-ndx 17140 df-base 17156 df-plusg 17209 df-0g 17380 df-gsum 17381 df-mgm 18543 df-sgrp 18622 df-mnd 18638

This theorem is referenced by: (None)

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