signlem0

Description: Adding a zero as the highest coefficient does not change the parity of the sign changes. (Contributed by Thierry Arnoux, 12-Oct-2018)

	Ref	Expression
Hypotheses	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 ↦ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
Assertion	signlem0	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( 𝑉 ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) = ( 𝑉 ‘ 𝐹 ) )

Proof

Step	Hyp	Ref	Expression
1		signsv.p	⊢ ⨣ = ( 𝑎 ∈ { - 1 , 0 , 1 } , 𝑏 ∈ { - 1 , 0 , 1 } ↦ if ( 𝑏 = 0 , 𝑎 , 𝑏 ) )
2		signsv.w	⊢ 𝑊 = { ⟨ ( Base ‘ ndx ) , { - 1 , 0 , 1 } ⟩ , ⟨ ( +_g ‘ ndx ) , ⨣ ⟩ }
3		signsv.t	⊢ 𝑇 = ( 𝑓 ∈ Word ℝ ↦ ( 𝑛 ∈ ( 0 ..^ ( ♯ ‘ 𝑓 ) ) ↦ ( 𝑊 Σ_g ( 𝑖 ∈ ( 0 ... 𝑛 ) ↦ ( sgn ‘ ( 𝑓 ‘ 𝑖 ) ) ) ) ) )
4		signsv.v	⊢ 𝑉 = ( 𝑓 ∈ Word ℝ ↦ Σ 𝑗 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) if ( ( ( 𝑇 ‘ 𝑓 ) ‘ 𝑗 ) ≠ ( ( 𝑇 ‘ 𝑓 ) ‘ ( 𝑗 − 1 ) ) , 1 , 0 ) )
5		0re	⊢ 0 ∈ ℝ
6	1 2 3 4	signsvfn	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ 0 ∈ ℝ ) → ( 𝑉 ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) = ( ( 𝑉 ‘ 𝐹 ) + if ( ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) < 0 , 1 , 0 ) ) )
7	5 6	mpan2	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( 𝑉 ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) = ( ( 𝑉 ‘ 𝐹 ) + if ( ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) < 0 , 1 , 0 ) ) )
8	5	ltnri	⊢ ¬ 0 < 0
9		neg1cn	⊢ - 1 ∈ ℂ
10		ax-1cn	⊢ 1 ∈ ℂ
11		prssi	⊢ ( ( - 1 ∈ ℂ ∧ 1 ∈ ℂ ) → { - 1 , 1 } ⊆ ℂ )
12	9 10 11	mp2an	⊢ { - 1 , 1 } ⊆ ℂ
13		simpl	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) )
14		eldifsn	⊢ ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ↔ ( 𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅ ) )
15	13 14	sylib	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( 𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅ ) )
16		lennncl	⊢ ( ( 𝐹 ∈ Word ℝ ∧ 𝐹 ≠ ∅ ) → ( ♯ ‘ 𝐹 ) ∈ ℕ )
17		fzo0end	⊢ ( ( ♯ ‘ 𝐹 ) ∈ ℕ → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
18	15 16 17	3syl	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) )
19	1 2 3 4	signstfvcl	⊢ ( ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) ∧ ( ( ♯ ‘ 𝐹 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ∈ { - 1 , 1 } )
20	18 19	mpdan	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ∈ { - 1 , 1 } )
21	12 20	sselid	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) ∈ ℂ )
22	21	mul01d	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) = 0 )
23	22	breq1d	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) < 0 ↔ 0 < 0 ) )
24	8 23	mtbiri	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ¬ ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) < 0 )
25	24	iffalsed	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → if ( ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) < 0 , 1 , 0 ) = 0 )
26	25	oveq2d	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( 𝑉 ‘ 𝐹 ) + if ( ( ( ( 𝑇 ‘ 𝐹 ) ‘ ( ( ♯ ‘ 𝐹 ) − 1 ) ) · 0 ) < 0 , 1 , 0 ) ) = ( ( 𝑉 ‘ 𝐹 ) + 0 ) )
27	1 2 3 4	signsvvf	⊢ 𝑉 : Word ℝ ⟶ ℕ₀
28	27	a1i	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → 𝑉 : Word ℝ ⟶ ℕ₀ )
29	13	eldifad	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → 𝐹 ∈ Word ℝ )
30	28 29	ffvelcdmd	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( 𝑉 ‘ 𝐹 ) ∈ ℕ₀ )
31	30	nn0cnd	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( 𝑉 ‘ 𝐹 ) ∈ ℂ )
32	31	addridd	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( ( 𝑉 ‘ 𝐹 ) + 0 ) = ( 𝑉 ‘ 𝐹 ) )
33	7 26 32	3eqtrd	⊢ ( ( 𝐹 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐹 ‘ 0 ) ≠ 0 ) → ( 𝑉 ‘ ( 𝐹 ++ ⟨“ 0 ”⟩ ) ) = ( 𝑉 ‘ 𝐹 ) )

Metamath Proof Explorer

Theorem signlem0

Proof