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Theorem signstfvn 34258
Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 8-Oct-2018.)
Hypotheses
Ref Expression
signsv.p ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
signsv.w π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
signsv.t 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
signsv.v 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(β™―β€˜π‘“))if(((π‘‡β€˜π‘“)β€˜π‘—) β‰  ((π‘‡β€˜π‘“)β€˜(𝑗 βˆ’ 1)), 1, 0))
Assertion
Ref Expression
signstfvn ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((π‘‡β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))β€˜(β™―β€˜πΉ)) = (((π‘‡β€˜πΉ)β€˜((β™―β€˜πΉ) βˆ’ 1)) ⨣ (sgnβ€˜πΎ)))
Distinct variable groups:   π‘Ž,𝑏, ⨣   𝑓,𝑖,𝑛,𝐹   𝑓,𝐾,𝑖,𝑛   𝑓,π‘Š,𝑖,𝑛
Allowed substitution hints:   ⨣ (𝑓,𝑖,𝑗,𝑛)   𝑇(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   𝐹(𝑗,π‘Ž,𝑏)   𝐾(𝑗,π‘Ž,𝑏)   𝑉(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   π‘Š(𝑗,π‘Ž,𝑏)

Proof of Theorem signstfvn
StepHypRef Expression
1 signsv.p . . . . 5 ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
2 signsv.w . . . . 5 π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
31, 2signswbase 34243 . . . 4 {-1, 0, 1} = (Baseβ€˜π‘Š)
41, 2signswmnd 34246 . . . . 5 π‘Š ∈ Mnd
54a1i 11 . . . 4 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ π‘Š ∈ Mnd)
6 eldifi 4119 . . . . . . . . 9 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ 𝐹 ∈ Word ℝ)
7 lencl 14515 . . . . . . . . 9 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„•0)
86, 7syl 17 . . . . . . . 8 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ (β™―β€˜πΉ) ∈ β„•0)
9 eldifsn 4786 . . . . . . . . 9 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ↔ (𝐹 ∈ Word ℝ ∧ 𝐹 β‰  βˆ…))
10 hasheq0 14354 . . . . . . . . . . 11 (𝐹 ∈ Word ℝ β†’ ((β™―β€˜πΉ) = 0 ↔ 𝐹 = βˆ…))
1110necon3bid 2975 . . . . . . . . . 10 (𝐹 ∈ Word ℝ β†’ ((β™―β€˜πΉ) β‰  0 ↔ 𝐹 β‰  βˆ…))
1211biimpar 476 . . . . . . . . 9 ((𝐹 ∈ Word ℝ ∧ 𝐹 β‰  βˆ…) β†’ (β™―β€˜πΉ) β‰  0)
139, 12sylbi 216 . . . . . . . 8 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ (β™―β€˜πΉ) β‰  0)
14 elnnne0 12516 . . . . . . . 8 ((β™―β€˜πΉ) ∈ β„• ↔ ((β™―β€˜πΉ) ∈ β„•0 ∧ (β™―β€˜πΉ) β‰  0))
158, 13, 14sylanbrc 581 . . . . . . 7 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ (β™―β€˜πΉ) ∈ β„•)
1615adantr 479 . . . . . 6 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (β™―β€˜πΉ) ∈ β„•)
17 nnm1nn0 12543 . . . . . 6 ((β™―β€˜πΉ) ∈ β„• β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0)
1816, 17syl 17 . . . . 5 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ β„•0)
19 nn0uz 12894 . . . . 5 β„•0 = (β„€β‰₯β€˜0)
2018, 19eleqtrdi 2835 . . . 4 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (β„€β‰₯β€˜0))
21 ccatws1cl 14598 . . . . . . . . . 10 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (𝐹 ++ βŸ¨β€œπΎβ€βŸ©) ∈ Word ℝ)
2221adantr 479 . . . . . . . . 9 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ (𝐹 ++ βŸ¨β€œπΎβ€βŸ©) ∈ Word ℝ)
23 wrdf 14501 . . . . . . . . 9 ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©) ∈ Word ℝ β†’ (𝐹 ++ βŸ¨β€œπΎβ€βŸ©):(0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©)))βŸΆβ„)
2422, 23syl 17 . . . . . . . 8 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ (𝐹 ++ βŸ¨β€œπΎβ€βŸ©):(0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©)))βŸΆβ„)
257nn0zd 12614 . . . . . . . . . . . . 13 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„€)
26 fzoval 13665 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) ∈ β„€ β†’ (0..^(β™―β€˜πΉ)) = (0...((β™―β€˜πΉ) βˆ’ 1)))
2725, 26syl 17 . . . . . . . . . . . 12 (𝐹 ∈ Word ℝ β†’ (0..^(β™―β€˜πΉ)) = (0...((β™―β€˜πΉ) βˆ’ 1)))
2827adantr 479 . . . . . . . . . . 11 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0..^(β™―β€˜πΉ)) = (0...((β™―β€˜πΉ) βˆ’ 1)))
29 fzossfz 13683 . . . . . . . . . . 11 (0..^(β™―β€˜πΉ)) βŠ† (0...(β™―β€˜πΉ))
3028, 29eqsstrrdi 4028 . . . . . . . . . 10 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0...(β™―β€˜πΉ)))
31 s1cl 14584 . . . . . . . . . . . . . 14 (𝐾 ∈ ℝ β†’ βŸ¨β€œπΎβ€βŸ© ∈ Word ℝ)
32 ccatlen 14557 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ βŸ¨β€œπΎβ€βŸ© ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©)) = ((β™―β€˜πΉ) + (β™―β€˜βŸ¨β€œπΎβ€βŸ©)))
3331, 32sylan2 591 . . . . . . . . . . . . 13 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©)) = ((β™―β€˜πΉ) + (β™―β€˜βŸ¨β€œπΎβ€βŸ©)))
34 s1len 14588 . . . . . . . . . . . . . 14 (β™―β€˜βŸ¨β€œπΎβ€βŸ©) = 1
3534oveq2i 7427 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) + (β™―β€˜βŸ¨β€œπΎβ€βŸ©)) = ((β™―β€˜πΉ) + 1)
3633, 35eqtrdi 2781 . . . . . . . . . . . 12 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©)) = ((β™―β€˜πΉ) + 1))
3736oveq2d 7432 . . . . . . . . . . 11 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))) = (0..^((β™―β€˜πΉ) + 1)))
3825adantr 479 . . . . . . . . . . . . 13 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (β™―β€˜πΉ) ∈ β„€)
3938peano2zd 12699 . . . . . . . . . . . 12 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ ((β™―β€˜πΉ) + 1) ∈ β„€)
40 fzoval 13665 . . . . . . . . . . . 12 (((β™―β€˜πΉ) + 1) ∈ β„€ β†’ (0..^((β™―β€˜πΉ) + 1)) = (0...(((β™―β€˜πΉ) + 1) βˆ’ 1)))
4139, 40syl 17 . . . . . . . . . . 11 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0..^((β™―β€˜πΉ) + 1)) = (0...(((β™―β€˜πΉ) + 1) βˆ’ 1)))
427nn0cnd 12564 . . . . . . . . . . . . . 14 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„‚)
43 1cnd 11239 . . . . . . . . . . . . . 14 (𝐹 ∈ Word ℝ β†’ 1 ∈ β„‚)
4442, 43pncand 11602 . . . . . . . . . . . . 13 (𝐹 ∈ Word ℝ β†’ (((β™―β€˜πΉ) + 1) βˆ’ 1) = (β™―β€˜πΉ))
4544adantr 479 . . . . . . . . . . . 12 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (((β™―β€˜πΉ) + 1) βˆ’ 1) = (β™―β€˜πΉ))
4645oveq2d 7432 . . . . . . . . . . 11 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0...(((β™―β€˜πΉ) + 1) βˆ’ 1)) = (0...(β™―β€˜πΉ)))
4737, 41, 463eqtrd 2769 . . . . . . . . . 10 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))) = (0...(β™―β€˜πΉ)))
4830, 47sseqtrrd 4014 . . . . . . . . 9 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (0...((β™―β€˜πΉ) βˆ’ 1)) βŠ† (0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))))
4948sselda 3972 . . . . . . . 8 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ 𝑖 ∈ (0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))))
5024, 49ffvelcdmd 7090 . . . . . . 7 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) ∈ ℝ)
516, 50sylanl1 678 . . . . . 6 (((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) ∈ ℝ)
5251rexrd 11294 . . . . 5 (((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) ∈ ℝ*)
53 sgncl 34215 . . . . 5 (((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) ∈ ℝ* β†’ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)) ∈ {-1, 0, 1})
5452, 53syl 17 . . . 4 (((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)) ∈ {-1, 0, 1})
551, 2signswplusg 34244 . . . 4 ⨣ = (+gβ€˜π‘Š)
56 rexr 11290 . . . . . 6 (𝐾 ∈ ℝ β†’ 𝐾 ∈ ℝ*)
57 sgncl 34215 . . . . . 6 (𝐾 ∈ ℝ* β†’ (sgnβ€˜πΎ) ∈ {-1, 0, 1})
5856, 57syl 17 . . . . 5 (𝐾 ∈ ℝ β†’ (sgnβ€˜πΎ) ∈ {-1, 0, 1})
5958adantl 480 . . . 4 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (sgnβ€˜πΎ) ∈ {-1, 0, 1})
60 id 22 . . . . . . . . 9 (𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1) β†’ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1))
6142, 43npcand 11605 . . . . . . . . . 10 (𝐹 ∈ Word ℝ β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
6261adantr 479 . . . . . . . . 9 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
6360, 62sylan9eqr 2787 . . . . . . . 8 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1)) β†’ 𝑖 = (β™―β€˜πΉ))
6463fveq2d 6896 . . . . . . 7 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1)) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) = ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜(β™―β€˜πΉ)))
65 ccatws1ls 14615 . . . . . . . 8 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜(β™―β€˜πΉ)) = 𝐾)
6665adantr 479 . . . . . . 7 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1)) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜(β™―β€˜πΉ)) = 𝐾)
6764, 66eqtrd 2765 . . . . . 6 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1)) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) = 𝐾)
686, 67sylanl1 678 . . . . 5 (((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1)) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) = 𝐾)
6968fveq2d 6896 . . . 4 (((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) ∧ 𝑖 = (((β™―β€˜πΉ) βˆ’ 1) + 1)) β†’ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)) = (sgnβ€˜πΎ))
703, 5, 20, 54, 55, 59, 69gsumnunsn 34230 . . 3 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (π‘Š Ξ£g (𝑖 ∈ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))) = ((π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))) ⨣ (sgnβ€˜πΎ)))
716, 61syl 17 . . . . . . 7 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
7271adantr 479 . . . . . 6 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (((β™―β€˜πΉ) βˆ’ 1) + 1) = (β™―β€˜πΉ))
7372oveq2d 7432 . . . . 5 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) = (0...(β™―β€˜πΉ)))
7473mpteq1d 5238 . . . 4 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (𝑖 ∈ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–))) = (𝑖 ∈ (0...(β™―β€˜πΉ)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–))))
7574oveq2d 7432 . . 3 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (π‘Š Ξ£g (𝑖 ∈ (0...(((β™―β€˜πΉ) βˆ’ 1) + 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))) = (π‘Š Ξ£g (𝑖 ∈ (0...(β™―β€˜πΉ)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))))
76 simpll 765 . . . . . . . . 9 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ 𝐹 ∈ Word ℝ)
7731ad2antlr 725 . . . . . . . . 9 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ βŸ¨β€œπΎβ€βŸ© ∈ Word ℝ)
7828eleq2d 2811 . . . . . . . . . 10 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (𝑖 ∈ (0..^(β™―β€˜πΉ)) ↔ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))))
7978biimpar 476 . . . . . . . . 9 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ 𝑖 ∈ (0..^(β™―β€˜πΉ)))
80 ccatval1 14559 . . . . . . . . 9 ((𝐹 ∈ Word ℝ ∧ βŸ¨β€œπΎβ€βŸ© ∈ Word ℝ ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) = (πΉβ€˜π‘–))
8176, 77, 79, 80syl3anc 1368 . . . . . . . 8 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ ((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–) = (πΉβ€˜π‘–))
8281fveq2d 6896 . . . . . . 7 (((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) ∧ 𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1))) β†’ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)) = (sgnβ€˜(πΉβ€˜π‘–)))
8382mpteq2dva 5243 . . . . . 6 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–))) = (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–))))
846, 83sylan 578 . . . . 5 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–))) = (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–))))
8584oveq2d 7432 . . . 4 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))) = (π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))))
8685oveq1d 7431 . . 3 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))) ⨣ (sgnβ€˜πΎ)) = ((π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))) ⨣ (sgnβ€˜πΎ)))
8770, 75, 863eqtr3d 2773 . 2 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (π‘Š Ξ£g (𝑖 ∈ (0...(β™―β€˜πΉ)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))) = ((π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))) ⨣ (sgnβ€˜πΎ)))
88 eqid 2725 . . . . . . . 8 (β™―β€˜πΉ) = (β™―β€˜πΉ)
8988olci 864 . . . . . . 7 ((β™―β€˜πΉ) ∈ (0..^(β™―β€˜πΉ)) ∨ (β™―β€˜πΉ) = (β™―β€˜πΉ))
907, 19eleqtrdi 2835 . . . . . . . 8 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ (β„€β‰₯β€˜0))
91 fzosplitsni 13775 . . . . . . . 8 ((β™―β€˜πΉ) ∈ (β„€β‰₯β€˜0) β†’ ((β™―β€˜πΉ) ∈ (0..^((β™―β€˜πΉ) + 1)) ↔ ((β™―β€˜πΉ) ∈ (0..^(β™―β€˜πΉ)) ∨ (β™―β€˜πΉ) = (β™―β€˜πΉ))))
9290, 91syl 17 . . . . . . 7 (𝐹 ∈ Word ℝ β†’ ((β™―β€˜πΉ) ∈ (0..^((β™―β€˜πΉ) + 1)) ↔ ((β™―β€˜πΉ) ∈ (0..^(β™―β€˜πΉ)) ∨ (β™―β€˜πΉ) = (β™―β€˜πΉ))))
9389, 92mpbiri 257 . . . . . 6 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ (0..^((β™―β€˜πΉ) + 1)))
9493adantr 479 . . . . 5 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (β™―β€˜πΉ) ∈ (0..^((β™―β€˜πΉ) + 1)))
9594, 37eleqtrrd 2828 . . . 4 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ (β™―β€˜πΉ) ∈ (0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))))
96 signsv.t . . . . 5 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
97 signsv.v . . . . 5 𝑉 = (𝑓 ∈ Word ℝ ↦ Σ𝑗 ∈ (1..^(β™―β€˜π‘“))if(((π‘‡β€˜π‘“)β€˜π‘—) β‰  ((π‘‡β€˜π‘“)β€˜(𝑗 βˆ’ 1)), 1, 0))
981, 2, 96, 97signstfval 34253 . . . 4 (((𝐹 ++ βŸ¨β€œπΎβ€βŸ©) ∈ Word ℝ ∧ (β™―β€˜πΉ) ∈ (0..^(β™―β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©)))) β†’ ((π‘‡β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))β€˜(β™―β€˜πΉ)) = (π‘Š Ξ£g (𝑖 ∈ (0...(β™―β€˜πΉ)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))))
9921, 95, 98syl2anc 582 . . 3 ((𝐹 ∈ Word ℝ ∧ 𝐾 ∈ ℝ) β†’ ((π‘‡β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))β€˜(β™―β€˜πΉ)) = (π‘Š Ξ£g (𝑖 ∈ (0...(β™―β€˜πΉ)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))))
1006, 99sylan 578 . 2 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((π‘‡β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))β€˜(β™―β€˜πΉ)) = (π‘Š Ξ£g (𝑖 ∈ (0...(β™―β€˜πΉ)) ↦ (sgnβ€˜((𝐹 ++ βŸ¨β€œπΎβ€βŸ©)β€˜π‘–)))))
101 fzo0end 13756 . . . . . 6 ((β™―β€˜πΉ) ∈ β„• β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
10215, 101syl 17 . . . . 5 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ)))
1031, 2, 96, 97signstfval 34253 . . . . 5 ((𝐹 ∈ Word ℝ ∧ ((β™―β€˜πΉ) βˆ’ 1) ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜πΉ)β€˜((β™―β€˜πΉ) βˆ’ 1)) = (π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))))
1046, 102, 103syl2anc 582 . . . 4 (𝐹 ∈ (Word ℝ βˆ– {βˆ…}) β†’ ((π‘‡β€˜πΉ)β€˜((β™―β€˜πΉ) βˆ’ 1)) = (π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))))
105104adantr 479 . . 3 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((π‘‡β€˜πΉ)β€˜((β™―β€˜πΉ) βˆ’ 1)) = (π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))))
106105oveq1d 7431 . 2 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ (((π‘‡β€˜πΉ)β€˜((β™―β€˜πΉ) βˆ’ 1)) ⨣ (sgnβ€˜πΎ)) = ((π‘Š Ξ£g (𝑖 ∈ (0...((β™―β€˜πΉ) βˆ’ 1)) ↦ (sgnβ€˜(πΉβ€˜π‘–)))) ⨣ (sgnβ€˜πΎ)))
10787, 100, 1063eqtr4d 2775 1 ((𝐹 ∈ (Word ℝ βˆ– {βˆ…}) ∧ 𝐾 ∈ ℝ) β†’ ((π‘‡β€˜(𝐹 ++ βŸ¨β€œπΎβ€βŸ©))β€˜(β™―β€˜πΉ)) = (((π‘‡β€˜πΉ)β€˜((β™―β€˜πΉ) βˆ’ 1)) ⨣ (sgnβ€˜πΎ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1533   ∈ wcel 2098   β‰  wne 2930   βˆ– cdif 3936  βˆ…c0 4318  ifcif 4524  {csn 4624  {cpr 4626  {ctp 4628  βŸ¨cop 4630   ↦ cmpt 5226  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  β„cr 11137  0cc0 11138  1c1 11139   + caddc 11141  β„*cxr 11277   βˆ’ cmin 11474  -cneg 11475  β„•cn 12242  β„•0cn0 12502  β„€cz 12588  β„€β‰₯cuz 12852  ...cfz 13516  ..^cfzo 13659  β™―chash 14321  Word cword 14496   ++ cconcat 14552  βŸ¨β€œcs1 14577  sgncsgn 15065  Ξ£csu 15664  ndxcnx 17161  Basecbs 17179  +gcplusg 17232   Ξ£g cgsu 17421  Mndcmnd 18693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-seq 13999  df-hash 14322  df-word 14497  df-concat 14553  df-s1 14578  df-sgn 15066  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17180  df-plusg 17245  df-0g 17422  df-gsum 17423  df-mgm 18599  df-sgrp 18678  df-mnd 18694
This theorem is referenced by:  signsvtn0  34259  signstfvneq0  34261  signstfveq0  34266  signsvfn  34271
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