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Theorem signstfvc 33883
Description: Zero-skipping sign in a word compared to a shorter word. (Contributed by Thierry Arnoux, 11-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
signstfvc ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
Distinct variable groups:   π‘Ž,𝑏, ⨣   𝑓,𝑖,𝑛,𝐹   𝑓,π‘Š,𝑖,𝑛   𝑖,𝑁,𝑛
Allowed substitution hints:   ⨣ (𝑓,𝑖,𝑗,𝑛)   𝑇(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   𝐹(𝑗,π‘Ž,𝑏)   𝐺(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   𝑁(𝑓,𝑗,π‘Ž,𝑏)   𝑉(𝑓,𝑖,𝑗,𝑛,π‘Ž,𝑏)   π‘Š(𝑗,π‘Ž,𝑏)

Proof of Theorem signstfvc
Dummy variables 𝑒 𝑔 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7419 . . . . . . . 8 (𝑔 = βˆ… β†’ (𝐹 ++ 𝑔) = (𝐹 ++ βˆ…))
21fveq2d 6894 . . . . . . 7 (𝑔 = βˆ… β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ βˆ…)))
32fveq1d 6892 . . . . . 6 (𝑔 = βˆ… β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘))
43eqeq1d 2732 . . . . 5 (𝑔 = βˆ… β†’ (((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) ↔ ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)))
54imbi2d 339 . . . 4 (𝑔 = βˆ… β†’ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))))
6 oveq2 7419 . . . . . . . 8 (𝑔 = 𝑒 β†’ (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
76fveq2d 6894 . . . . . . 7 (𝑔 = 𝑒 β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ 𝑒)))
87fveq1d 6892 . . . . . 6 (𝑔 = 𝑒 β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘))
98eqeq1d 2732 . . . . 5 (𝑔 = 𝑒 β†’ (((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) ↔ ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)))
109imbi2d 339 . . . 4 (𝑔 = 𝑒 β†’ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))))
11 oveq2 7419 . . . . . . . 8 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))
1211fveq2d 6894 . . . . . . 7 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©))))
1312fveq1d 6892 . . . . . 6 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘))
1413eqeq1d 2732 . . . . 5 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ (((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) ↔ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)))
1514imbi2d 339 . . . 4 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))))
16 oveq2 7419 . . . . . . . 8 (𝑔 = 𝐺 β†’ (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
1716fveq2d 6894 . . . . . . 7 (𝑔 = 𝐺 β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ 𝐺)))
1817fveq1d 6892 . . . . . 6 (𝑔 = 𝐺 β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘))
1918eqeq1d 2732 . . . . 5 (𝑔 = 𝐺 β†’ (((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) ↔ ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)))
2019imbi2d 339 . . . 4 (𝑔 = 𝐺 β†’ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)) ↔ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))))
21 ccatrid 14541 . . . . . . 7 (𝐹 ∈ Word ℝ β†’ (𝐹 ++ βˆ…) = 𝐹)
2221fveq2d 6894 . . . . . 6 (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜(𝐹 ++ βˆ…)) = (π‘‡β€˜πΉ))
2322fveq1d 6892 . . . . 5 (𝐹 ∈ Word ℝ β†’ ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
2423adantr 479 . . . 4 ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
25 s1cl 14556 . . . . . . . . . . . . . 14 (π‘˜ ∈ ℝ β†’ βŸ¨β€œπ‘˜β€βŸ© ∈ Word ℝ)
26 ccatass 14542 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ βŸ¨β€œπ‘˜β€βŸ© ∈ Word ℝ) β†’ ((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©) = (𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))
2725, 26syl3an3 1163 . . . . . . . . . . . . 13 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ) β†’ ((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©) = (𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))
28273expb 1118 . . . . . . . . . . . 12 ((𝐹 ∈ Word ℝ ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ ((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©) = (𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))
2928adantlr 711 . . . . . . . . . . 11 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ ((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©) = (𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))
3029fveq2d 6894 . . . . . . . . . 10 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ (π‘‡β€˜((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©)) = (π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©))))
3130fveq1d 6892 . . . . . . . . 9 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ ((π‘‡β€˜((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘))
32 ccatcl 14528 . . . . . . . . . . 11 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (𝐹 ++ 𝑒) ∈ Word ℝ)
3332ad2ant2r 743 . . . . . . . . . 10 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ (𝐹 ++ 𝑒) ∈ Word ℝ)
34 simprr 769 . . . . . . . . . 10 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ π‘˜ ∈ ℝ)
35 lencl 14487 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„•0)
3635nn0zd 12588 . . . . . . . . . . . . . . 15 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„€)
3736adantr 479 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜πΉ) ∈ β„€)
38 lencl 14487 . . . . . . . . . . . . . . . 16 ((𝐹 ++ 𝑒) ∈ Word ℝ β†’ (β™―β€˜(𝐹 ++ 𝑒)) ∈ β„•0)
3932, 38syl 17 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) ∈ β„•0)
4039nn0zd 12588 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) ∈ β„€)
4135nn0red 12537 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ ℝ)
42 lencl 14487 . . . . . . . . . . . . . . . 16 (𝑒 ∈ Word ℝ β†’ (β™―β€˜π‘’) ∈ β„•0)
43 nn0addge1 12522 . . . . . . . . . . . . . . . 16 (((β™―β€˜πΉ) ∈ ℝ ∧ (β™―β€˜π‘’) ∈ β„•0) β†’ (β™―β€˜πΉ) ≀ ((β™―β€˜πΉ) + (β™―β€˜π‘’)))
4441, 42, 43syl2an 594 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜πΉ) ≀ ((β™―β€˜πΉ) + (β™―β€˜π‘’)))
45 ccatlen 14529 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) = ((β™―β€˜πΉ) + (β™―β€˜π‘’)))
4644, 45breqtrrd 5175 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜(𝐹 ++ 𝑒)))
47 eluz2 12832 . . . . . . . . . . . . . 14 ((β™―β€˜(𝐹 ++ 𝑒)) ∈ (β„€β‰₯β€˜(β™―β€˜πΉ)) ↔ ((β™―β€˜πΉ) ∈ β„€ ∧ (β™―β€˜(𝐹 ++ 𝑒)) ∈ β„€ ∧ (β™―β€˜πΉ) ≀ (β™―β€˜(𝐹 ++ 𝑒))))
4837, 40, 46, 47syl3anbrc 1341 . . . . . . . . . . . . 13 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) ∈ (β„€β‰₯β€˜(β™―β€˜πΉ)))
49 fzoss2 13664 . . . . . . . . . . . . 13 ((β™―β€˜(𝐹 ++ 𝑒)) ∈ (β„€β‰₯β€˜(β™―β€˜πΉ)) β†’ (0..^(β™―β€˜πΉ)) βŠ† (0..^(β™―β€˜(𝐹 ++ 𝑒))))
5048, 49syl 17 . . . . . . . . . . . 12 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (0..^(β™―β€˜πΉ)) βŠ† (0..^(β™―β€˜(𝐹 ++ 𝑒))))
5150ad2ant2r 743 . . . . . . . . . . 11 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ (0..^(β™―β€˜πΉ)) βŠ† (0..^(β™―β€˜(𝐹 ++ 𝑒))))
52 simplr 765 . . . . . . . . . . 11 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ 𝑁 ∈ (0..^(β™―β€˜πΉ)))
5351, 52sseldd 3982 . . . . . . . . . 10 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ 𝑁 ∈ (0..^(β™―β€˜(𝐹 ++ 𝑒))))
54 signsv.p . . . . . . . . . . 11 ⨣ = (π‘Ž ∈ {-1, 0, 1}, 𝑏 ∈ {-1, 0, 1} ↦ if(𝑏 = 0, π‘Ž, 𝑏))
55 signsv.w . . . . . . . . . . 11 π‘Š = {⟨(Baseβ€˜ndx), {-1, 0, 1}⟩, ⟨(+gβ€˜ndx), ⨣ ⟩}
56 signsv.t . . . . . . . . . . 11 𝑇 = (𝑓 ∈ Word ℝ ↦ (𝑛 ∈ (0..^(β™―β€˜π‘“)) ↦ (π‘Š Ξ£g (𝑖 ∈ (0...𝑛) ↦ (sgnβ€˜(π‘“β€˜π‘–))))))
57 signsv.v . . . . . . . . . . 11 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))
5854, 55, 56, 57signstfvp 33880 . . . . . . . . . 10 (((𝐹 ++ 𝑒) ∈ Word ℝ ∧ π‘˜ ∈ ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜(𝐹 ++ 𝑒)))) β†’ ((π‘‡β€˜((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘))
5933, 34, 53, 58syl3anc 1369 . . . . . . . . 9 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ ((π‘‡β€˜((𝐹 ++ 𝑒) ++ βŸ¨β€œπ‘˜β€βŸ©))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘))
6031, 59eqtr3d 2772 . . . . . . . 8 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘))
61 id 22 . . . . . . . 8 (((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) β†’ ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
6260, 61sylan9eq 2790 . . . . . . 7 ((((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) ∧ ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)) β†’ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
6362ex 411 . . . . . 6 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ (((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) β†’ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)))
6463expcom 412 . . . . 5 ((𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ) β†’ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ (((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘) β†’ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))))
6564a2d 29 . . . 4 ((𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ) β†’ (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)) β†’ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))))
665, 10, 15, 20, 24, 65wrdind 14676 . . 3 (𝐺 ∈ Word ℝ β†’ ((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘)))
67663impib 1114 . 2 ((𝐺 ∈ Word ℝ ∧ 𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
68673com12 1121 1 ((𝐹 ∈ Word ℝ ∧ 𝐺 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) β†’ ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938   βŠ† wss 3947  βˆ…c0 4321  ifcif 4527  {cpr 4629  {ctp 4631  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   ≀ cle 11253   βˆ’ cmin 11448  -cneg 11449  β„•0cn0 12476  β„€cz 12562  β„€β‰₯cuz 12826  ...cfz 13488  ..^cfzo 13631  β™―chash 14294  Word cword 14468   ++ cconcat 14524  βŸ¨β€œcs1 14549  sgncsgn 15037  Ξ£csu 15636  ndxcnx 17130  Basecbs 17148  +gcplusg 17201   Ξ£g cgsu 17390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525  df-s1 14550  df-substr 14595  df-pfx 14625
This theorem is referenced by:  signstres  33884
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