signstfvc - Mathbox for Thierry Arnoux

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Theorem signstfvc 31954

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 7143	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹 ++ 𝑔) = (𝐹 ++ ∅))
2	1	fveq2d 6649	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ ∅)))
3	2	fveq1d 6647	. . . . . 6 ⊢ (𝑔 = ∅ → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ ∅))‘𝑁))
4	3	eqeq1d 2800	. . . . 5 ⊢ (𝑔 = ∅ → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
5	4	imbi2d 344	. . . 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 7143	. . . . . . . 8 ⊢ (𝑔 = 𝑒 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
7	6	fveq2d 6649	. . . . . . 7 ⊢ (𝑔 = 𝑒 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝑒)))
8	7	fveq1d 6647	. . . . . 6 ⊢ (𝑔 = 𝑒 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
9	8	eqeq1d 2800	. . . . 5 ⊢ (𝑔 = 𝑒 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
10	9	imbi2d 344	. . . 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 7143	. . . . . . . 8 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
12	11	fveq2d 6649	. . . . . . 7 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
13	12	fveq1d 6647	. . . . . 6 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
14	13	eqeq1d 2800	. . . . 5 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
15	14	imbi2d 344	. . . 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 7143	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
17	16	fveq2d 6649	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝐺)))
18	17	fveq1d 6647	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝐺))‘𝑁))
19	18	eqeq1d 2800	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
20	19	imbi2d 344	. . . 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 13932	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ ∅) = 𝐹)
22	21	fveq2d 6649	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘(𝐹 ++ ∅)) = (𝑇‘𝐹))
23	22	fveq1d 6647	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
24	23	adantr 484	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
25		s1cl 13947	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℝ → ⟨“𝑘”⟩ ∈ Word ℝ)
26		ccatass 13933	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ ⟨“𝑘”⟩ ∈ Word ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
27	25, 26	syl3an3 1162	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
28	27	3expb 1117	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
29	28	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
30	29	fveq2d 6649	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
31	30	fveq1d 6647	. . . . . . . . 9 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
32		ccatcl 13917	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (𝐹 ++ 𝑒) ∈ Word ℝ)
33	32	ad2ant2r 746	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (𝐹 ++ 𝑒) ∈ Word ℝ)
34		simprr 772	. . . . . . . . . 10 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑘 ∈ ℝ)
35		lencl 13876	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
36	35	nn0zd 12073	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℤ)
37	36	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ∈ ℤ)
38		lencl 13876	. . . . . . . . . . . . . . . 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 12073	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ ℤ)
41	35	nn0red 11944	. . . . . . . . . . . . . . . 16 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℝ)
42		lencl 13876	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ Word ℝ → (♯‘𝑒) ∈ ℕ₀)
43		nn0addge1 11931	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐹) ∈ ℝ ∧ (♯‘𝑒) ∈ ℕ₀) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
44	41, 42, 43	syl2an 598	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
45		ccatlen 13918	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒)))
46	44, 45	breqtrrd 5058	. . . . . . . . . . . . . 14 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒)))
47		eluz2 12237	. . . . . . . . . . . . . 14 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) ↔ ((♯‘𝐹) ∈ ℤ ∧ (♯‘(𝐹 ++ 𝑒)) ∈ ℤ ∧ (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒))))
48	37, 40, 46, 47	syl3anbrc 1340	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)))
49		fzoss2 13060	. . . . . . . . . . . . 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 746	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
52		simplr 768	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) ∧ (𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ)) → 𝑁 ∈ (0..^(♯‘𝐹)))
53	51, 52	sseldd 3916	. . . . . . . . . 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 31951	. . . . . . . . . 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 1368	. . . . . . . . 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 2835	. . . . . . . 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 2853	. . . . . . 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 416	. . . . . 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 417	. . . . 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 14075	. . 3 ⊢ (𝐺 ∈ Word ℝ → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
67	66	3impib 1113	. 2 ⊢ ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
68	67	3com12 1120	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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ifcif 4425 {cpr 4527 {ctp 4529 ⟨cop 4531 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 ≤ cle 10665 − cmin 10859 -cneg 10860 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ...cfz 12885 ..^cfzo 13028 ♯chash 13686 Word cword 13857 ++ cconcat 13913 ⟨“cs1 13940 sgncsgn 14437 Σcsu 15034 ndxcnx 16472 Basecbs 16475 +_gcplusg 16557 Σ_g cgsu 16706

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029 df-hash 13687 df-word 13858 df-lsw 13906 df-concat 13914 df-s1 13941 df-substr 13994 df-pfx 14024

This theorem is referenced by: signstres 31955

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