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Theorem signstfvc 33585
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 7417 . . . . . . . 8 (𝑔 = βˆ… β†’ (𝐹 ++ 𝑔) = (𝐹 ++ βˆ…))
21fveq2d 6896 . . . . . . 7 (𝑔 = βˆ… β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ βˆ…)))
32fveq1d 6894 . . . . . 6 (𝑔 = βˆ… β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘))
43eqeq1d 2735 . . . . 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 341 . . . 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 7417 . . . . . . . 8 (𝑔 = 𝑒 β†’ (𝐹 ++ 𝑔) = (𝐹 ++ 𝑒))
76fveq2d 6896 . . . . . . 7 (𝑔 = 𝑒 β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ 𝑒)))
87fveq1d 6894 . . . . . 6 (𝑔 = 𝑒 β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝑒))β€˜π‘))
98eqeq1d 2735 . . . . 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 341 . . . 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 7417 . . . . . . . 8 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ (𝐹 ++ 𝑔) = (𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©)))
1211fveq2d 6896 . . . . . . 7 (𝑔 = (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©) β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ (𝑒 ++ βŸ¨β€œπ‘˜β€βŸ©))))
1312fveq1d 6894 . . . . . 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 2735 . . . . 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 341 . . . 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 7417 . . . . . . . 8 (𝑔 = 𝐺 β†’ (𝐹 ++ 𝑔) = (𝐹 ++ 𝐺))
1716fveq2d 6896 . . . . . . 7 (𝑔 = 𝐺 β†’ (π‘‡β€˜(𝐹 ++ 𝑔)) = (π‘‡β€˜(𝐹 ++ 𝐺)))
1817fveq1d 6894 . . . . . 6 (𝑔 = 𝐺 β†’ ((π‘‡β€˜(𝐹 ++ 𝑔))β€˜π‘) = ((π‘‡β€˜(𝐹 ++ 𝐺))β€˜π‘))
1918eqeq1d 2735 . . . . 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 341 . . . 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 14537 . . . . . . 7 (𝐹 ∈ Word ℝ β†’ (𝐹 ++ βˆ…) = 𝐹)
2221fveq2d 6896 . . . . . 6 (𝐹 ∈ Word ℝ β†’ (π‘‡β€˜(𝐹 ++ βˆ…)) = (π‘‡β€˜πΉ))
2322fveq1d 6894 . . . . 5 (𝐹 ∈ Word ℝ β†’ ((π‘‡β€˜(𝐹 ++ βˆ…))β€˜π‘) = ((π‘‡β€˜πΉ)β€˜π‘))
2423adantr 482 . . . 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 14552 . . . . . . . . . . . . . 14 (π‘˜ ∈ ℝ β†’ βŸ¨β€œπ‘˜β€βŸ© ∈ Word ℝ)
26 ccatass 14538 . . . . . . . . . . . . . 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 1166 . . . . . . . . . . . . 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 1121 . . . . . . . . . . . 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 714 . . . . . . . . . . 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 6896 . . . . . . . . . 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 6894 . . . . . . . . 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 14524 . . . . . . . . . . 11 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (𝐹 ++ 𝑒) ∈ Word ℝ)
3332ad2ant2r 746 . . . . . . . . . 10 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ (𝐹 ++ 𝑒) ∈ Word ℝ)
34 simprr 772 . . . . . . . . . 10 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ π‘˜ ∈ ℝ)
35 lencl 14483 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„•0)
3635nn0zd 12584 . . . . . . . . . . . . . . 15 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ β„€)
3736adantr 482 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜πΉ) ∈ β„€)
38 lencl 14483 . . . . . . . . . . . . . . . 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 12584 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) ∈ β„€)
4135nn0red 12533 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word ℝ β†’ (β™―β€˜πΉ) ∈ ℝ)
42 lencl 14483 . . . . . . . . . . . . . . . 16 (𝑒 ∈ Word ℝ β†’ (β™―β€˜π‘’) ∈ β„•0)
43 nn0addge1 12518 . . . . . . . . . . . . . . . 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 597 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜πΉ) ≀ ((β™―β€˜πΉ) + (β™―β€˜π‘’)))
45 ccatlen 14525 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) = ((β™―β€˜πΉ) + (β™―β€˜π‘’)))
4644, 45breqtrrd 5177 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜πΉ) ≀ (β™―β€˜(𝐹 ++ 𝑒)))
47 eluz2 12828 . . . . . . . . . . . . . 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 1344 . . . . . . . . . . . . 13 ((𝐹 ∈ Word ℝ ∧ 𝑒 ∈ Word ℝ) β†’ (β™―β€˜(𝐹 ++ 𝑒)) ∈ (β„€β‰₯β€˜(β™―β€˜πΉ)))
49 fzoss2 13660 . . . . . . . . . . . . 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 746 . . . . . . . . . . 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 768 . . . . . . . . . . 11 (((𝐹 ∈ Word ℝ ∧ 𝑁 ∈ (0..^(β™―β€˜πΉ))) ∧ (𝑒 ∈ Word ℝ ∧ π‘˜ ∈ ℝ)) β†’ 𝑁 ∈ (0..^(β™―β€˜πΉ)))
5351, 52sseldd 3984 . . . . . . . . . 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 33582 . . . . . . . . . 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 1372 . . . . . . . . 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 2775 . . . . . . . 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 2793 . . . . . . 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 414 . . . . . 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 415 . . . . 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 14672 . . 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 1117 . 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 1124 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941   βŠ† wss 3949  βˆ…c0 4323  ifcif 4529  {cpr 4631  {ctp 4633  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  -cneg 11445  β„•0cn0 12472  β„€cz 12558  β„€β‰₯cuz 12822  ...cfz 13484  ..^cfzo 13627  β™―chash 14290  Word cword 14464   ++ cconcat 14520  βŸ¨β€œcs1 14545  sgncsgn 15033  Ξ£csu 15632  ndxcnx 17126  Basecbs 17144  +gcplusg 17197   Ξ£g cgsu 17386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-substr 14591  df-pfx 14621
This theorem is referenced by:  signstres  33586
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