signstfvc - Mathbox for Thierry Arnoux

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Theorem signstfvc 31846

Description: Zero-skipping sign in a word compared to a shorter word. (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
signstfvc	⊢ ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))

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

Proof of Theorem signstfvc Dummy variables 𝑒 𝑔 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7166	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹 ++ 𝑔) = (𝐹 ++ ∅))
2	1	fveq2d 6676	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ ∅)))
3	2	fveq1d 6674	. . . . . 6 ⊢ (𝑔 = ∅ → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ ∅))‘𝑁))
4	3	eqeq1d 2825	. . . . 5 ⊢ (𝑔 = ∅ → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
5	4	imbi2d 343	. . . 4 ⊢ (𝑔 = ∅ → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))))
6		oveq2 7166	. . . . . . . 8 ⊢ (𝑔 = 𝑒 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
7	6	fveq2d 6676	. . . . . . 7 ⊢ (𝑔 = 𝑒 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝑒)))
8	7	fveq1d 6674	. . . . . 6 ⊢ (𝑔 = 𝑒 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
9	8	eqeq1d 2825	. . . . 5 ⊢ (𝑔 = 𝑒 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
10	9	imbi2d 343	. . . 4 ⊢ (𝑔 = 𝑒 → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁))))
11		oveq2 7166	. . . . . . . 8 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
12	11	fveq2d 6676	. . . . . . 7 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
13	12	fveq1d 6674	. . . . . 6 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
14	13	eqeq1d 2825	. . . . 5 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
15	14	imbi2d 343	. . . 4 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁))))
16		oveq2 7166	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
17	16	fveq2d 6676	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝐺)))
18	17	fveq1d 6674	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝐺))‘𝑁))
19	18	eqeq1d 2825	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
20	19	imbi2d 343	. . . 4 ⊢ (𝑔 = 𝐺 → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))))
21		ccatrid 13943	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ ∅) = 𝐹)
22	21	fveq2d 6676	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘(𝐹 ++ ∅)) = (𝑇‘𝐹))
23	22	fveq1d 6674	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
24	23	adantr 483	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
25		s1cl 13958	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → ⟨“𝑘”⟩ ∈ Word ℝ)
26		ccatass 13944	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ ⟨“𝑘”⟩ ∈ Word ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
27	25, 26	syl3an3 1161	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
28	27	3expb 1116	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
29	28	adantlr 713	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
30	29	fveq2d 6676	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
31	30	fveq1d 6674	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
32		ccatcl 13928	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (𝐹 ++ 𝑒) ∈ Word ℝ)
33	32	ad2ant2r 745	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝐹 ++ 𝑒) ∈ Word ℝ)
34		simprr 771	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑘 ∈ ℝ)
35		lencl 13885	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
36	35	nn0zd 12088	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
37	36	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ∈ ℤ)
38		lencl 13885	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ++ 𝑒) ∈ Word ℝ → (♯‘(𝐹 ++ 𝑒)) ∈ ℕ₀)
39	32, 38	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ ℕ₀)
40	39	nn0zd 12088	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ ℤ)
41	35	nn0red 11959	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℝ)
42		lencl 13885	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ Word ℝ → (♯‘𝑒) ∈ ℕ₀)
43		nn0addge1 11946	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℝ ∧ (♯‘𝑒) ∈ ℕ₀) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
44	41, 42, 43	syl2an 597	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
45		ccatlen 13929	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒)))
46	44, 45	breqtrrd 5096	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒)))
47		eluz2 12252	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) ↔ ((♯‘𝐹) ∈ ℤ ∧ (♯‘(𝐹 ++ 𝑒)) ∈ ℤ ∧ (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒))))
48	37, 40, 46, 47	syl3anbrc 1339	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)))
49		fzoss2 13068	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
50	48, 49	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
51	50	ad2ant2r 745	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
52		simplr 767	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑁 ∈ (0..^(♯‘𝐹)))
53	51, 52	sseldd 3970	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑁 ∈ (0..^(♯‘(𝐹 ++ 𝑒))))
54		signsv.p	. . . . . . . . . . 11 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
55		signsv.w	. . . . . . . . . . 11 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
56		signsv.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
57		signsv.v	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
58	54, 55, 56, 57	signstfvp 31843	. . . . . . . . . 10 ⊢ (((𝐹 ++ 𝑒) ∈ Word ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘(𝐹 ++ 𝑒)))) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
59	33, 34, 53, 58	syl3anc 1367	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
60	31, 59	eqtr3d 2860	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
61		id 22	. . . . . . . 8 ⊢ (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
62	60, 61	sylan9eq 2878	. . . . . . 7 ⊢ ((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) ∧ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
63	62	ex 415	. . . . . 6 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
64	63	expcom 416	. . . . 5 ⊢ ((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁))))
65	64	a2d 29	. . . 4 ⊢ ((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁))))
66	5, 10, 15, 20, 24, 65	wrdind 14086	. . 3 ⊢ (𝐺 ∈ Word ℝ → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
67	66	3impib 1112	. 2 ⊢ ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
68	67	3com12 1119	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ⊆ wss 3938 ∅c0 4293 ifcif 4469 {cpr 4571 {ctp 4573 ⟨cop 4575 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 ≤ cle 10678 − cmin 10872 -cneg 10873 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 ++ cconcat 13924 ⟨“cs1 13951 sgncsgn 14447 Σcsu 15044 ndxcnx 16482 Basecbs 16485 +_gcplusg 16567 Σ_g cgsu 16716

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-xnn0 11971 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-lsw 13917 df-concat 13925 df-s1 13952 df-substr 14005 df-pfx 14035

This theorem is referenced by: signstres 31847

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