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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itcovalt2 Structured version   Visualization version   GIF version
Theorem itcovalt2 47363
Description: The value of the function that returns the n-th iterate of the "times 2 plus a constant" function with regard to composition. (Contributed by AV, 7-May-2024.)
Hypothesis
Ref Expression
itcovalt2.f 𝐹 = (𝑛 ∈ β„•0 ↦ ((2 Β· 𝑛) + 𝐢))
Assertion
Ref Expression
itcovalt2 ((𝐼 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢)))
Distinct variable groups:   𝐢,𝑛   𝑛,𝐼
Allowed substitution hint:   𝐹(𝑛)
Proof of Theorem itcovalt2
Dummy variables 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . 5 (π‘₯ = 0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜0))
2 oveq2 7417 . . . . . . . 8 (π‘₯ = 0 β†’ (2↑π‘₯) = (2↑0))
32oveq2d 7425 . . . . . . 7 (π‘₯ = 0 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑0)))
43oveq1d 7424 . . . . . 6 (π‘₯ = 0 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢))
54mpteq2dv 5251 . . . . 5 (π‘₯ = 0 β†’ (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢)))
61, 5eqeq12d 2749 . . . 4 (π‘₯ = 0 β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) ↔ ((IterCompβ€˜πΉ)β€˜0) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢))))
76imbi2d 341 . . 3 (π‘₯ = 0 β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢))) ↔ (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜0) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢)))))
8 fveq2 6892 . . . . 5 (π‘₯ = 𝑦 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜π‘¦))
9 oveq2 7417 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (2↑π‘₯) = (2↑𝑦))
109oveq2d 7425 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑𝑦)))
1110oveq1d 7424 . . . . . 6 (π‘₯ = 𝑦 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))
1211mpteq2dv 5251 . . . . 5 (π‘₯ = 𝑦 β†’ (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢)))
138, 12eqeq12d 2749 . . . 4 (π‘₯ = 𝑦 β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) ↔ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))))
1413imbi2d 341 . . 3 (π‘₯ = 𝑦 β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢))) ↔ (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢)))))
15 fveq2 6892 . . . . 5 (π‘₯ = (𝑦 + 1) β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)))
16 oveq2 7417 . . . . . . . 8 (π‘₯ = (𝑦 + 1) β†’ (2↑π‘₯) = (2↑(𝑦 + 1)))
1716oveq2d 7425 . . . . . . 7 (π‘₯ = (𝑦 + 1) β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))))
1817oveq1d 7424 . . . . . 6 (π‘₯ = (𝑦 + 1) β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢))
1918mpteq2dv 5251 . . . . 5 (π‘₯ = (𝑦 + 1) β†’ (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢)))
2015, 19eqeq12d 2749 . . . 4 (π‘₯ = (𝑦 + 1) β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) ↔ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢))))
2120imbi2d 341 . . 3 (π‘₯ = (𝑦 + 1) β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢))) ↔ (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢)))))
22 fveq2 6892 . . . . 5 (π‘₯ = 𝐼 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜πΌ))
23 oveq2 7417 . . . . . . . 8 (π‘₯ = 𝐼 β†’ (2↑π‘₯) = (2↑𝐼))
2423oveq2d 7425 . . . . . . 7 (π‘₯ = 𝐼 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑𝐼)))
2524oveq1d 7424 . . . . . 6 (π‘₯ = 𝐼 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢))
2625mpteq2dv 5251 . . . . 5 (π‘₯ = 𝐼 β†’ (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢)))
2722, 26eqeq12d 2749 . . . 4 (π‘₯ = 𝐼 β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢)) ↔ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢))))
2827imbi2d 341 . . 3 (π‘₯ = 𝐼 β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢))) ↔ (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢)))))
29 itcovalt2.f . . . 4 𝐹 = (𝑛 ∈ β„•0 ↦ ((2 Β· 𝑛) + 𝐢))
3029itcovalt2lem1 47361 . . 3 (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜0) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢)))
31 pm2.27 42 . . . . . . 7 (𝐢 ∈ β„•0 β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))) β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))))
3231adantl 483 . . . . . 6 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))) β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))))
3329itcovalt2lem2 47362 . . . . . 6 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢)) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢))))
3432, 33syld 47 . . . . 5 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢))))
3534ex 414 . . . 4 (𝑦 ∈ β„•0 β†’ (𝐢 ∈ β„•0 β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢)))))
3635com23 86 . . 3 (𝑦 ∈ β„•0 β†’ ((𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))) β†’ (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢)))))
377, 14, 21, 28, 30, 36nn0ind 12657 . 2 (𝐼 ∈ β„•0 β†’ (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢))))
3837imp 408 1 ((𝐼 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113   Β· cmul 11115   βˆ’ cmin 11444  2c2 12267  β„•0cn0 12472  β†‘cexp 14027  IterCompcitco 47343
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-inf2 9636  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-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-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  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-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-seq 13967  df-exp 14028  df-itco 47345
This theorem is referenced by:  ackval3  47369
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