signstfvc - Mathbox for Thierry Arnoux

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Theorem signstfvc 30985

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 6800	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹 ++ 𝑔) = (𝐹 ++ ∅))
2	1	fveq2d 6336	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ ∅)))
3	2	fveq1d 6334	. . . . . 6 ⊢ (𝑔 = ∅ → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ ∅))‘𝑁))
4	3	eqeq1d 2772	. . . . 5 ⊢ (𝑔 = ∅ → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
5	4	imbi2d 329	. . . 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 6800	. . . . . . . 8 ⊢ (𝑔 = 𝑒 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
7	6	fveq2d 6336	. . . . . . 7 ⊢ (𝑔 = 𝑒 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝑒)))
8	7	fveq1d 6334	. . . . . 6 ⊢ (𝑔 = 𝑒 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
9	8	eqeq1d 2772	. . . . 5 ⊢ (𝑔 = 𝑒 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
10	9	imbi2d 329	. . . 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 6800	. . . . . . . 8 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
12	11	fveq2d 6336	. . . . . . 7 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
13	12	fveq1d 6334	. . . . . 6 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
14	13	eqeq1d 2772	. . . . 5 ⊢ (𝑔 = (𝑒 ++ ⟨“𝑘”⟩) → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
15	14	imbi2d 329	. . . 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 6800	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
17	16	fveq2d 6336	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (𝑇‘(𝐹 ++ 𝑔)) = (𝑇‘(𝐹 ++ 𝐺)))
18	17	fveq1d 6334	. . . . . 6 ⊢ (𝑔 = 𝐺 → ((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘(𝐹 ++ 𝐺))‘𝑁))
19	18	eqeq1d 2772	. . . . 5 ⊢ (𝑔 = 𝐺 → (((𝑇‘(𝐹 ++ 𝑔))‘𝑁) = ((𝑇‘𝐹)‘𝑁) ↔ ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
20	19	imbi2d 329	. . . 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 13568	. . . . . . 7 ⊢ (𝐹 ∈ Word ℝ → (𝐹 ++ ∅) = 𝐹)
22	21	fveq2d 6336	. . . . . 6 ⊢ (𝐹 ∈ Word ℝ → (𝑇‘(𝐹 ++ ∅)) = (𝑇‘𝐹))
23	22	fveq1d 6334	. . . . 5 ⊢ (𝐹 ∈ Word ℝ → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
24	23	adantr 466	. . . 4 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ ∅))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
25		simprl 746	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → 𝐹 ∈ Word ℝ)
26		simpll 742	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → 𝑒 ∈ Word ℝ)
27		simplr 744	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → 𝑘 ∈ ℝ)
28	27	s1cld 13582	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → ⟨“𝑘”⟩ ∈ Word ℝ)
29		ccatass 13569	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ ⟨“𝑘”⟩ ∈ Word ℝ) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
30	25, 26, 28, 29	syl3anc 1475	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → ((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩) = (𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))
31	30	fveq2d 6336	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩)) = (𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩))))
32	31	fveq1d 6334	. . . . . . . . 9 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁))
33		ccatcl 13555	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (𝐹 ++ 𝑒) ∈ Word ℝ)
34	25, 26, 33	syl2anc 565	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (𝐹 ++ 𝑒) ∈ Word ℝ)
35		lencl 13519	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ Word ℝ → (♯‘𝐹) ∈ ℕ₀)
36	25, 35	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘𝐹) ∈ ℕ₀)
37	36	nn0zd 11681	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘𝐹) ∈ ℤ)
38		lencl 13519	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ++ 𝑒) ∈ Word ℝ → (♯‘(𝐹 ++ 𝑒)) ∈ ℕ₀)
39	34, 38	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘(𝐹 ++ 𝑒)) ∈ ℕ₀)
40	39	nn0zd 11681	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘(𝐹 ++ 𝑒)) ∈ ℤ)
41	36	nn0red 11553	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘𝐹) ∈ ℝ)
42		lencl 13519	. . . . . . . . . . . . . . . 16 ⊢ (𝑒 ∈ Word ℝ → (♯‘𝑒) ∈ ℕ₀)
43	26, 42	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘𝑒) ∈ ℕ₀)
44		nn0addge1 11540	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐹) ∈ ℝ ∧ (♯‘𝑒) ∈ ℕ₀) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
45	41, 43, 44	syl2anc 565	. . . . . . . . . . . . . 14 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘𝐹) ≤ ((♯‘𝐹) + (♯‘𝑒)))
46		ccatlen 13556	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) → (♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒)))
47	25, 26, 46	syl2anc 565	. . . . . . . . . . . . . 14 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘(𝐹 ++ 𝑒)) = ((♯‘𝐹) + (♯‘𝑒)))
48	45, 47	breqtrrd 4812	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒)))
49		eluz2 11893	. . . . . . . . . . . . 13 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) ↔ ((♯‘𝐹) ∈ ℤ ∧ (♯‘(𝐹 ++ 𝑒)) ∈ ℤ ∧ (♯‘𝐹) ≤ (♯‘(𝐹 ++ 𝑒))))
50	37, 40, 48, 49	syl3anbrc 1427	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)))
51		fzoss2 12703	. . . . . . . . . . . 12 ⊢ ((♯‘(𝐹 ++ 𝑒)) ∈ (ℤ_≥‘(♯‘𝐹)) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
52	50, 51	syl 17	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → (0..^(♯‘𝐹)) ⊆ (0..^(♯‘(𝐹 ++ 𝑒))))
53		simprr 748	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → 𝑁 ∈ (0..^(♯‘𝐹)))
54	52, 53	sseldd 3751	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → 𝑁 ∈ (0..^(♯‘(𝐹 ++ 𝑒))))
55		signsv.p	. . . . . . . . . . 11 ⊢ ⨣ = (𝑎 ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, 𝑎, 𝑏))
56		signsv.w	. . . . . . . . . . 11 ⊢ 𝑊 = {⟨(Base‘ndx), {-1, 0, 1}⟩, ⟨(+_g‘ndx), ⨣ ⟩}
57		signsv.t	. . . . . . . . . . 11 ⊢ 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(♯‘𝑓)) ↦ (𝑊 Σ_g (𝑖 ∈ (0...𝑛) ↦ (sgn‘(𝑓‘𝑖))))))
58		signsv.v	. . . . . . . . . . 11 ⊢ 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(♯‘𝑓))if(((𝑇‘𝑓)‘𝑗) ≠ ((𝑇‘𝑓)‘(𝑗 − 1)), 1, 0))
59	55, 56, 57, 58	signstfvp 30982	. . . . . . . . . 10 ⊢ (((𝐹 ++ 𝑒) ∈ Word ℝ ∧ 𝑘 ∈ ℝ ∧ 𝑁 ∈ (0..^(♯‘(𝐹 ++ 𝑒)))) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
60	34, 27, 54, 59	syl3anc 1475	. . . . . . . . 9 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → ((𝑇‘((𝐹 ++ 𝑒) ++ ⟨“𝑘”⟩))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
61	32, 60	eqtr3d 2806	. . . . . . . 8 ⊢ (((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) → ((𝑇‘(𝐹 ++ (𝑒 ++ ⟨“𝑘”⟩)))‘𝑁) = ((𝑇‘(𝐹 ++ 𝑒))‘𝑁))
62	61	adantr 466	. . . . . . 7 ⊢ ((((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 ++ 𝑒))‘𝑁))
63		simpr 471	. . . . . . 7 ⊢ ((((𝑒 ∈ Word ℝ ∧ 𝑘 ∈ ℝ) ∧ (𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹)))) ∧ ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁)) → ((𝑇‘(𝐹 ++ 𝑒))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
64	62, 63	eqtrd 2804	. . . . . 6 ⊢ ((((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	exp31 406	. . . . 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))))
66	65	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))))
67	5, 10, 15, 20, 24, 66	wrdind 13684	. . 3 ⊢ (𝐺 ∈ Word ℝ → ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁)))
68	67	3impib 1107	. 2 ⊢ ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(♯‘𝐹))) → ((𝑇‘(𝐹 ++ 𝐺))‘𝑁) = ((𝑇‘𝐹)‘𝑁))
69	68	3com12 1116	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 382 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 ≠ wne 2942 ⊆ wss 3721 ∅c0 4061 ifcif 4223 {cpr 4316 {ctp 4318 ⟨cop 4320 class class class wbr 4784 ↦ cmpt 4861 ‘cfv 6031 (class class class)co 6792 ↦ cmpt2 6794 ℝcr 10136 0cc0 10137 1c1 10138 + caddc 10140 ≤ cle 10276 − cmin 10467 -cneg 10468 ℕ₀cn0 11493 ℤcz 11578 ℤ_≥cuz 11887 ...cfz 12532 ..^cfzo 12672 ♯chash 13320 Word cword 13486 ++ cconcat 13488 ⟨“cs1 13489 sgncsgn 14033 Σcsu 14623 ndxcnx 16060 Basecbs 16063 +_gcplusg 16148 Σ_g cgsu 16308

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-card 8964 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-n0 11494 df-xnn0 11565 df-z 11579 df-uz 11888 df-fz 12533 df-fzo 12673 df-hash 13321 df-word 13494 df-lsw 13495 df-concat 13496 df-s1 13497 df-substr 13498

This theorem is referenced by: signstres 30986

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