signsvtp

Description: Adding a letter of the same sign as the highest coefficient does not change the sign. (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 ) )
	signsvf.e	⊢ ( 𝜑 → 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) )
	signsvf.0	⊢ ( 𝜑 → ( 𝐸 ‘ 0 ) ≠ 0 )
	signsvf.f	⊢ ( 𝜑 → 𝐹 = ( 𝐸 ++ ⟨“ 𝐴 ”⟩ ) )
	signsvf.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	signsvf.n	⊢ 𝑁 = ( ♯ ‘ 𝐸 )
	signsvt.b	⊢ 𝐵 = ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) )
Assertion	signsvtp	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝑉 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐸 ) )

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		signsvf.e	⊢ ( 𝜑 → 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) )
6		signsvf.0	⊢ ( 𝜑 → ( 𝐸 ‘ 0 ) ≠ 0 )
7		signsvf.f	⊢ ( 𝜑 → 𝐹 = ( 𝐸 ++ ⟨“ 𝐴 ”⟩ ) )
8		signsvf.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
9		signsvf.n	⊢ 𝑁 = ( ♯ ‘ 𝐸 )
10		signsvt.b	⊢ 𝐵 = ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) )
11	7	fveq2d	⊢ ( 𝜑 → ( 𝑉 ‘ 𝐹 ) = ( 𝑉 ‘ ( 𝐸 ++ ⟨“ 𝐴 ”⟩ ) ) )
12	1 2 3 4	signsvfn	⊢ ( ( ( 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ∧ ( 𝐸 ‘ 0 ) ≠ 0 ) ∧ 𝐴 ∈ ℝ ) → ( 𝑉 ‘ ( 𝐸 ++ ⟨“ 𝐴 ”⟩ ) ) = ( ( 𝑉 ‘ 𝐸 ) + if ( ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 , 1 , 0 ) ) )
13	5 6 8 12	syl21anc	⊢ ( 𝜑 → ( 𝑉 ‘ ( 𝐸 ++ ⟨“ 𝐴 ”⟩ ) ) = ( ( 𝑉 ‘ 𝐸 ) + if ( ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 , 1 , 0 ) ) )
14	11 13	eqtrd	⊢ ( 𝜑 → ( 𝑉 ‘ 𝐹 ) = ( ( 𝑉 ‘ 𝐸 ) + if ( ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 , 1 , 0 ) ) )
15	14	adantr	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝑉 ‘ 𝐹 ) = ( ( 𝑉 ‘ 𝐸 ) + if ( ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 , 1 , 0 ) ) )
16		0red	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 ∈ ℝ )
17	5	adantr	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) )
18	17	eldifad	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝐸 ∈ Word ℝ )
19	1 2 3 4	signstf	⊢ ( 𝐸 ∈ Word ℝ → ( 𝑇 ‘ 𝐸 ) ∈ Word ℝ )
20		wrdf	⊢ ( ( 𝑇 ‘ 𝐸 ) ∈ Word ℝ → ( 𝑇 ‘ 𝐸 ) : ( 0 ..^ ( ♯ ‘ ( 𝑇 ‘ 𝐸 ) ) ) ⟶ ℝ )
21	18 19 20	3syl	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝑇 ‘ 𝐸 ) : ( 0 ..^ ( ♯ ‘ ( 𝑇 ‘ 𝐸 ) ) ) ⟶ ℝ )
22		eldifsn	⊢ ( 𝐸 ∈ ( Word ℝ ∖ { ∅ } ) ↔ ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) )
23	5 22	sylib	⊢ ( 𝜑 → ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) )
24	23	adantr	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) )
25		lennncl	⊢ ( ( 𝐸 ∈ Word ℝ ∧ 𝐸 ≠ ∅ ) → ( ♯ ‘ 𝐸 ) ∈ ℕ )
26		fzo0end	⊢ ( ( ♯ ‘ 𝐸 ) ∈ ℕ → ( ( ♯ ‘ 𝐸 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) )
27	24 25 26	3syl	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ( ♯ ‘ 𝐸 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ 𝐸 ) ) )
28	1 2 3 4	signstlen	⊢ ( 𝐸 ∈ Word ℝ → ( ♯ ‘ ( 𝑇 ‘ 𝐸 ) ) = ( ♯ ‘ 𝐸 ) )
29	18 28	syl	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ♯ ‘ ( 𝑇 ‘ 𝐸 ) ) = ( ♯ ‘ 𝐸 ) )
30	29	oveq2d	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 0 ..^ ( ♯ ‘ ( 𝑇 ‘ 𝐸 ) ) ) = ( 0 ..^ ( ♯ ‘ 𝐸 ) ) )
31	27 30	eleqtrrd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ( ♯ ‘ 𝐸 ) − 1 ) ∈ ( 0 ..^ ( ♯ ‘ ( 𝑇 ‘ 𝐸 ) ) ) )
32	21 31	ffvelcdmd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) ∈ ℝ )
33	8	adantr	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝐴 ∈ ℝ )
34	32 33	remulcld	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) ∈ ℝ )
35		simpr	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 < ( 𝐴 · 𝐵 ) )
36	9	oveq1i	⊢ ( 𝑁 − 1 ) = ( ( ♯ ‘ 𝐸 ) − 1 )
37	36	fveq2i	⊢ ( ( 𝑇 ‘ 𝐸 ) ‘ ( 𝑁 − 1 ) ) = ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) )
38	10 37	eqtri	⊢ 𝐵 = ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) )
39	38 32	eqeltrid	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝐵 ∈ ℝ )
40	39	recnd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝐵 ∈ ℂ )
41	33	recnd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝐴 ∈ ℂ )
42	40 41	mulcomd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝐵 · 𝐴 ) = ( 𝐴 · 𝐵 ) )
43	35 42	breqtrrd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 < ( 𝐵 · 𝐴 ) )
44	38	oveq1i	⊢ ( 𝐵 · 𝐴 ) = ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 )
45	43 44	breqtrdi	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 0 < ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) )
46	16 34 45	ltnsymd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ¬ ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 )
47	46	iffalsed	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → if ( ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 , 1 , 0 ) = 0 )
48	47	oveq2d	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ( 𝑉 ‘ 𝐸 ) + if ( ( ( ( 𝑇 ‘ 𝐸 ) ‘ ( ( ♯ ‘ 𝐸 ) − 1 ) ) · 𝐴 ) < 0 , 1 , 0 ) ) = ( ( 𝑉 ‘ 𝐸 ) + 0 ) )
49	1 2 3 4	signsvvf	⊢ 𝑉 : Word ℝ ⟶ ℕ₀
50	49	a1i	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → 𝑉 : Word ℝ ⟶ ℕ₀ )
51	50 18	ffvelcdmd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝑉 ‘ 𝐸 ) ∈ ℕ₀ )
52	51	nn0cnd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝑉 ‘ 𝐸 ) ∈ ℂ )
53	52	addridd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( ( 𝑉 ‘ 𝐸 ) + 0 ) = ( 𝑉 ‘ 𝐸 ) )
54	15 48 53	3eqtrd	⊢ ( ( 𝜑 ∧ 0 < ( 𝐴 · 𝐵 ) ) → ( 𝑉 ‘ 𝐹 ) = ( 𝑉 ‘ 𝐸 ) )

Metamath Proof Explorer

Theorem signsvtp

Proof