signstres - Mathbox for Thierry Arnoux

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Theorem signstres 34607

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 34598	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ)
6		wrdf 14536	. . . . . . 7 ⊢ ((𝑇‘𝐹) ∈ Word ℝ → (𝑇‘𝐹):(0..^(♯‘(𝑇‘𝐹)))⟶ℝ)
7		ffn 6706	. . . . . . 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 34599	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(𝑇‘𝐹)) = (♯‘𝐹))
10	9	oveq2d 7421	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(𝑇‘𝐹))) = (0..^(♯‘𝐹)))
11	10	fneq2d 6632	. . . . . 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 32604	. . . . 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 13538	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ_≥‘𝑁))
17		fzoss2 13704	. . . . . 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 4184	. . . . . 6 ⊢ ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = ((0..^(♯‘𝐹)) ∩ (0..^𝑁))
21		dfss2 3944	. . . . . . 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 2784	. . . . 5 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) → ((0..^(♯‘𝐹)) ∩ (0..^𝑁)) = (0..^𝑁))
24	23	fneq2d 6632	. . . 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 32910	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ)
28	1, 2, 3, 4	signstf 34598	. . . 4 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ)
29		wrdf 14536	. . . 4 ⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ)
30		ffn 6706	. . . 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 34599	. . . . . . 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 14546	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 𝐹 Fn (0..^(♯‘𝐹)))
35		fnssres 6661	. . . . . . . 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 14393	. . . . . . 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 13637	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → 𝑁 ∈ ℕ₀)
40		hashfzo0 14448	. . . . . . . 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 2774	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = 𝑁)
44	43	oveq2d 7421	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) = (0..^𝑁))
45	44	fneq2d 6632	. . 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 6895	. . . . 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 6880	. . . . 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 6878	. . . 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 2770	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
55	54	oveq2d 7421	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁))
56	55	eleq2d 2820	. . . . . . 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 34606	. . . . 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 2774	. . 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 32911	. . . 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 3148	. 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 7024	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 2108 ≠ wne 2932 ∃wrex 3060 ∩ cin 3925 ⊆ wss 3926 ifcif 4500 {cpr 4603 {ctp 4605 ⟨cop 4607 ↦ cmpt 5201 ↾ cres 5656 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ∈ cmpo 7407 ℝcr 11128 0cc0 11129 1c1 11130 − cmin 11466 -cneg 11467 ℕ₀cn0 12501 ℤ_≥cuz 12852 ...cfz 13524 ..^cfzo 13671 ♯chash 14348 Word cword 14531 ++ cconcat 14588 sgncsgn 15105 Σcsu 15702 ndxcnx 17212 Basecbs 17228 +_gcplusg 17271 Σ_g cgsu 17454

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-n0 12502 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13525 df-fzo 13672 df-seq 14020 df-hash 14349 df-word 14532 df-lsw 14581 df-concat 14589 df-s1 14614 df-substr 14659 df-pfx 14689 df-sgn 15106 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-0g 17455 df-gsum 17456 df-mgm 18618 df-sgrp 18697 df-mnd 18713

This theorem is referenced by: (None)

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