signstfvc - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > signstfvc		Structured version Visualization version GIF version

Theorem signstfvc 32453

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 7263	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹 ++ 𝑔) = (𝐹 ++ ∅))
2	1	fveq2d 6760	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ ∅)))
3	2	fveq1d 6758	. . . . . 6 ⊢ (𝑔 = ∅ → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ ∅))‘𝑁))
4	3	eqeq1d 2740	. . . . 5 ⊢ (𝑔 = ∅ → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
5	4	imbi2d 340	. . . 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 7263	. . . . . . . 8 ⊢ (𝑔 = 𝑒 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
7	6	fveq2d 6760	. . . . . . 7 ⊢ (𝑔 = 𝑒 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝑒)))
8	7	fveq1d 6758	. . . . . 6 ⊢ (𝑔 = 𝑒 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
9	8	eqeq1d 2740	. . . . 5 ⊢ (𝑔 = 𝑒 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
10	9	imbi2d 340	. . . 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 7263	. . . . . . . 8 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
12	11	fveq2d 6760	. . . . . . 7 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
13	12	fveq1d 6758	. . . . . 6 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
14	13	eqeq1d 2740	. . . . 5 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
15	14	imbi2d 340	. . . 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 7263	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
17	16	fveq2d 6760	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝐺)))
18	17	fveq1d 6758	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝐺))‘𝑁))
19	18	eqeq1d 2740	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
20	19	imbi2d 340	. . . 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 14220	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ ∅) = 𝐹)
22	21	fveq2d 6760	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘(𝐹 ++ ∅)) = (𝑇‘𝐹))
23	22	fveq1d 6758	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
24	23	adantr 480	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
25		s1cl 14235	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → ⟨“𝑘”⟩ ∈ Word ℝ)
26		ccatass 14221	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ ⟨“𝑘”⟩ ∈ Word ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
27	25, 26	syl3an3 1163	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
28	27	3expb 1118	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
29	28	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
30	29	fveq2d 6760	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
31	30	fveq1d 6758	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
32		ccatcl 14205	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (𝐹 ++ 𝑒) ∈ Word ℝ)
33	32	ad2ant2r 743	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝐹 ++ 𝑒) ∈ Word ℝ)
34		simprr 769	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑘 ∈ ℝ)
35		lencl 14164	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
36	35	nn0zd 12353	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ∈ ℤ)
38		lencl 14164	. . . . . . . . . . . . . . . 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 12353	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ ℤ)
41	35	nn0red 12224	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℝ)
42		lencl 14164	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ Word ℝ → (♯‘𝑒) ∈ ℕ₀)
43		nn0addge1 12209	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℝ ∧ (♯‘𝑒) ∈ ℕ₀) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
44	41, 42, 43	syl2an 595	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
45		ccatlen 14206	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒)))
46	44, 45	breqtrrd 5098	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒)))
47		eluz2 12517	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) ↔ ((♯‘𝐹) ∈ ℤ ∧ (♯‘(𝐹 ++ 𝑒)) ∈ ℤ ∧ (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒))))
48	37, 40, 46, 47	syl3anbrc 1341	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)))
49		fzoss2 13343	. . . . . . . . . . . . 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 743	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
52		simplr 765	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑁 ∈ (0..^(♯‘𝐹)))
53	51, 52	sseldd 3918	. . . . . . . . . 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 32450	. . . . . . . . . 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 1369	. . . . . . . . 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 2780	. . . . . . . 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 2799	. . . . . . 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 412	. . . . . 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 413	. . . . 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 14363	. . 3 ⊢ (𝐺 ∈ Word ℝ → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
67	66	3impib 1114	. 2 ⊢ ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
68	67	3com12 1121	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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ⊆ wss 3883 ∅c0 4253 ifcif 4456 {cpr 4560 {ctp 4562 ⟨cop 4564 class class class wbr 5070 ↦ cmpt 5153 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ℝcr 10801 0cc0 10802 1c1 10803 + caddc 10805 ≤ cle 10941 − cmin 11135 -cneg 11136 ℕ₀cn0 12163 ℤcz 12249 ℤ_≥cuz 12511 ...cfz 13168 ..^cfzo 13311 ♯chash 13972 Word cword 14145 ++ cconcat 14201 ⟨“cs1 14228 sgncsgn 14725 Σcsu 15325 ndxcnx 16822 Basecbs 16840 +_gcplusg 16888 Σ_g cgsu 17068

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-xnn0 12236 df-z 12250 df-uz 12512 df-fz 13169 df-fzo 13312 df-hash 13973 df-word 14146 df-lsw 14194 df-concat 14202 df-s1 14229 df-substr 14282 df-pfx 14312

This theorem is referenced by: signstres 32454

W3C validator