signstfvc - Mathbox for Thierry Arnoux

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Theorem signstfvc 35090

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 7425	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹 ++ 𝑔) = (𝐹 ++ ∅))
2	1	fveq2d 6886	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ ∅)))
3	2	fveq1d 6884	. . . . . 6 ⊢ (𝑔 = ∅ → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ ∅))‘𝑁))
4	3	eqeq1d 2764	. . . . 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 7425	. . . . . . . 8 ⊢ (𝑔 = 𝑒 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
7	6	fveq2d 6886	. . . . . . 7 ⊢ (𝑔 = 𝑒 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝑒)))
8	7	fveq1d 6884	. . . . . 6 ⊢ (𝑔 = 𝑒 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
9	8	eqeq1d 2764	. . . . 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 7425	. . . . . . . 8 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
12	11	fveq2d 6886	. . . . . . 7 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
13	12	fveq1d 6884	. . . . . 6 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
14	13	eqeq1d 2764	. . . . 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 7425	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
17	16	fveq2d 6886	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝐺)))
18	17	fveq1d 6884	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝐺))‘𝑁))
19	18	eqeq1d 2764	. . . . 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 14657	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ ∅) = 𝐹)
22	21	fveq2d 6886	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘(𝐹 ++ ∅)) = (𝑇‘𝐹))
23	22	fveq1d 6884	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
24	23	adantr 486	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
25		s1cl 14673	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → ⟨“𝑘”⟩ ∈ Word ℝ)
26		ccatass 14658	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ ⟨“𝑘”⟩ ∈ Word ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
27	25, 26	syl3an3 1183	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
28	27	3expb 1138	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
29	28	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
30	29	fveq2d 6886	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
31	30	fveq1d 6884	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
32		ccatcl 14643	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (𝐹 ++ 𝑒) ∈ Word ℝ)
33	32	ad2ant2r 760	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝐹 ++ 𝑒) ∈ Word ℝ)
34		simprr 785	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑘 ∈ ℝ)
35		lencl 14602	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
36	35	nn0zd 12644	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
37	36	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ∈ ℤ)
38		lencl 14602	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 ++ 𝑒) ∈ Word ℝ → (♯‘(𝐹 ++ 𝑒)) ∈ ℕ₀)
39	32, 38	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ ℕ₀)
40	39	nn0zd 12644	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ ℤ)
41	35	nn0red 12594	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℝ)
42		lencl 14602	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ Word ℝ → (♯‘𝑒) ∈ ℕ₀)
43		nn0addge1 12578	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℝ ∧ (♯‘𝑒) ∈ ℕ₀) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
44	41, 42, 43	syl2an 608	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
45		ccatlen 14644	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒)))
46	44, 45	breqtrrd 5137	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒)))
47		eluz2 12897	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) ↔ ((♯‘𝐹) ∈ ℤ ∧ (♯‘(𝐹 ++ 𝑒)) ∈ ℤ ∧ (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒))))
48	37, 40, 46, 47	syl3anbrc 1362	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)))
49		fzoss2 13747	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
50	48, 49	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
51	50	ad2ant2r 760	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
52		simplr 781	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑁 ∈ (0..^(♯‘𝐹)))
53	51, 52	sseldd 3935	. . . . . . . . . 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 35087	. . . . . . . . . 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 1398	. . . . . . . . 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 2799	. . . . . . . 8 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
61		id 23	. . . . . . . 8 ⊢ (((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
62	60, 61	sylan9eq 2817	. . . . . . 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 418	. . . . . 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 419	. . . . 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 30	. . . 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 14795	. . 3 ⊢ (𝐺 ∈ Word ℝ → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
67	66	3impib 1134	. 2 ⊢ ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
68	67	3com12 1141	1 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ⊆ wss 3902 ∅c0 4282 ifcif 4485 {cpr 4589 {ctp 4591 ⟨cop 4593 class class class wbr 5107 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7417 ∈ cmpo 7419 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 ≤ cle 11272 − cmin 11469 -cneg 11470 ℕ₀cn0 12532 ℤcz 12619 ℤ_≥cuz 12891 ...cfz 13565 ..^cfzo 13713 ♯chash 14398 Word cword 14582 ++ cconcat 14639 ⟨“cs1 14666 sgncsgn 15163 Σcsu 15777 ndxcnx 17291 Basecbs 17307 +_gcplusg 17348 Σ_g cgsu 17531

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-n0 12533 df-xnn0 12606 df-z 12620 df-uz 12892 df-fz 13566 df-fzo 13714 df-hash 14399 df-word 14583 df-lsw 14632 df-concat 14640 df-s1 14667 df-substr 14713 df-pfx 14745

This theorem is used by: signstres 35091

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