signstres - Mathbox for Thierry Arnoux

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Theorem signstres 35251

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 35242	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) ∈ Word ℝ)
6		wrdf 14667	. . . . . . 7 ⊢ ((𝑇‘𝐹) ∈ Word ℝ → (𝑇‘𝐹):(0..^(♯‘(𝑇‘𝐹)))⟶ℝ)
7		ffn 6710	. . . . . . 7 ⊢ ((𝑇‘𝐹):(0..^(♯‘(𝑇‘𝐹)))⟶ℝ → (𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹))))
8	5, 6, 7	3syl 19	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹))))
9	1, 2, 3, 4	signstlen 35243	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → (♯‘(𝑇‘𝐹)) = (♯‘𝐹))
10	9	oveq2d 7437	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (0..^(♯‘(𝑇‘𝐹))) = (0..^(♯‘𝐹)))
11	10	fneq2d 6634	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘𝐹) Fn (0..^(♯‘(𝑇‘𝐹))) ↔ (𝑇‘𝐹) Fn (0..^(♯‘𝐹))))
12	8, 11	mpbid 235	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘𝐹) Fn (0..^(♯‘𝐹)))
13		fnresin 33266	. . . . 5 ⊢ ((𝑇‘𝐹) Fn (0..^(♯‘𝐹)) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁)))
14	12, 13	syl 18	. . . 4 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁)))
15	14	adantr 486	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn ((0..^(♯‘𝐹)) ∩ (0..^𝑁)))
16		elfzuz3 13657	. . . . . 6 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ_≥‘𝑁))
17		fzoss2 13826	. . . . . 6 ⊢ ((♯‘𝐹) ∈ (ℤ_≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
18	16, 17	syl 18	. . . . 5 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
19	18	adantl 487	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^𝑁) ⊆ (0..^(♯‘𝐹)))
20		incom 4106	. . . . . 6 ⊢ ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = ((0..^(♯‘𝐹)) ∩ (0..^𝑁))
21		dfss2 4094	. . . . . . 7 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) ↔ ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = (0..^𝑁))
22	21	biimpi 219	. . . . . 6 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) → ((0..^𝑁) ∩ (0..^(♯‘𝐹))) = (0..^𝑁))
23	20, 22	eqtr3id 2809	. . . . 5 ⊢ ((0..^𝑁) ⊆ (0..^(♯‘𝐹)) → ((0..^(♯‘𝐹)) ∩ (0..^𝑁)) = (0..^𝑁))
24	23	fneq2d 6634	. . . 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 18	. . 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 235	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) Fn (0..^𝑁))
27		wrdres 33549	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) ∈ Word ℝ)
28	1, 2, 3, 4	signstf 35242	. . . 4 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ)
29		wrdf 14667	. . . 4 ⊢ ((𝑇‘(𝐹 ↾ (0..^𝑁))) ∈ Word ℝ → (𝑇‘(𝐹 ↾ (0..^𝑁))):(0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))))⟶ℝ)
30		ffn 6710	. . . 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 20	. . 3 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁))))))
32	1, 2, 3, 4	signstlen 35243	. . . . . . 7 ⊢ ((𝐹 ↾ (0..^𝑁)) ∈ Word ℝ → (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = (♯‘(𝐹 ↾ (0..^𝑁))))
33	27, 32	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = (♯‘(𝐹 ↾ (0..^𝑁))))
34		wrdfn 14677	. . . . . . . 8 ⊢ (𝐹 ∈ Word ℝ → 𝐹 Fn (0..^(♯‘𝐹)))
35		fnssres 6663	. . . . . . . 8 ⊢ ((𝐹 Fn (0..^(♯‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
36	34, 18, 35	syl2an 608	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
37		hashfn 14523	. . . . . . 7 ⊢ ((𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁) → (♯‘(𝐹 ↾ (0..^𝑁))) = (♯‘(0..^𝑁)))
38	36, 37	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = (♯‘(0..^𝑁)))
39		elfznn0 13758	. . . . . . . 8 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → 𝑁 ∈ ℕ₀)
40		hashfzo0 14579	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(0..^𝑁)) = 𝑁)
41	39, 40	syl 18	. . . . . . 7 ⊢ (𝑁 ∈ (0...(♯‘𝐹)) → (♯‘(0..^𝑁)) = 𝑁)
42	41	adantl 487	. . . . . 6 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(0..^𝑁)) = 𝑁)
43	33, 38, 42	3eqtrd 2799	. . . . 5 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝑇‘(𝐹 ↾ (0..^𝑁)))) = 𝑁)
44	43	oveq2d 7437	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝑇‘(𝐹 ↾ (0..^𝑁))))) = (0..^𝑁))
45	44	fneq2d 6634	. . 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 235	. 2 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝑇‘(𝐹 ↾ (0..^𝑁))) Fn (0..^𝑁))
47		fvres 6905	. . . . 5 ⊢ (𝑚 ∈ (0..^𝑁) → (((𝑇‘𝐹) ↾ (0..^𝑁))‘𝑚) = ((𝑇‘𝐹)‘𝑚))
48	47	ad3antlr 744	. . . 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 490	. . . . . 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 6890	. . . . 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 6888	. . . 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 743	. . . . 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 781	. . . . 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 2795	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
55	54	oveq2d 7437	. . . . . . . 8 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (0..^(♯‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁))
56	55	eleq2d 2846	. . . . . . 7 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝑚 ∈ (0..^(♯‘(𝐹 ↾ (0..^𝑁)))) ↔ 𝑚 ∈ (0..^𝑁)))
57	56	biimpar 483	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → 𝑚 ∈ (0..^(♯‘(𝐹 ↾ (0..^𝑁)))))
58	57	ad2antrr 739	. . . . 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 35250	. . . . 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 1398	. . . 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 2799	. . 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 33550	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
63	62	adantr 486	. . 3 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) ∧ 𝑚 ∈ (0..^𝑁)) → ∃𝑔 ∈ Word ℝ𝐹 = ((𝐹 ↾ (0..^𝑁)) ++ 𝑔))
64	61, 63	r19.29a 3170	. 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 7033	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0...(♯‘𝐹))) → ((𝑇‘𝐹) ↾ (0..^𝑁)) = (𝑇‘(𝐹 ↾ (0..^𝑁))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∃wrex 3086 ⊆ wss 3895 ∩ cin 4089 ifcif 4481 {cpr 4586 {ctp 4588 ⟨cop 4834 ↦ cmpt 5185 ↾ cres 5652 Fn wfn 6534 ⟶wf 6535 ‘cfv 6539 (class class class)co 7421 ∈ cmpo 7423 ℝcr 11203 0cc0 11204 1c1 11205 − cmin 11545 -cneg 11546 ℕ₀cn0 12610 ℤ_≥cuz 12969 ...cfz 13643 ..^cfzo 13792 ♯chash 14478 Word cword 14662 ++ cconcat 14719 sgncsgn 15245 Σcsu 15859 ndxcnx 17377 Basecbs 17393 +_gcplusg 17434 Σ_g cgsu 17617

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-n0 12611 df-xnn0 12684 df-z 12698 df-uz 12970 df-fz 13644 df-fzo 13793 df-seq 14149 df-hash 14479 df-word 14663 df-lsw 14712 df-concat 14720 df-s1 14747 df-substr 14794 df-pfx 14826 df-sgn 15246 df-struct 17331 df-slot 17366 df-ndx 17378 df-base 17394 df-plusg 17447 df-0g 17618 df-gsum 17619 df-mgm 18822 df-sgrp 18914 df-mnd 18930

This theorem is used by: (None)

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