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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itcovalt2 Structured version   Visualization version   GIF version
Theorem itcovalt2 46916
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 6862 . . . . 5 (π‘₯ = 0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜0))
2 oveq2 7385 . . . . . . . 8 (π‘₯ = 0 β†’ (2↑π‘₯) = (2↑0))
32oveq2d 7393 . . . . . . 7 (π‘₯ = 0 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑0)))
43oveq1d 7392 . . . . . 6 (π‘₯ = 0 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢))
54mpteq2dv 5227 . . . . 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 2747 . . . 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 340 . . 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 6862 . . . . 5 (π‘₯ = 𝑦 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜π‘¦))
9 oveq2 7385 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (2↑π‘₯) = (2↑𝑦))
109oveq2d 7393 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑𝑦)))
1110oveq1d 7392 . . . . . 6 (π‘₯ = 𝑦 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))
1211mpteq2dv 5227 . . . . 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 2747 . . . 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 340 . . 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 6862 . . . . 5 (π‘₯ = (𝑦 + 1) β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)))
16 oveq2 7385 . . . . . . . 8 (π‘₯ = (𝑦 + 1) β†’ (2↑π‘₯) = (2↑(𝑦 + 1)))
1716oveq2d 7393 . . . . . . 7 (π‘₯ = (𝑦 + 1) β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))))
1817oveq1d 7392 . . . . . 6 (π‘₯ = (𝑦 + 1) β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢))
1918mpteq2dv 5227 . . . . 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 2747 . . . 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 340 . . 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 6862 . . . . 5 (π‘₯ = 𝐼 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜πΌ))
23 oveq2 7385 . . . . . . . 8 (π‘₯ = 𝐼 β†’ (2↑π‘₯) = (2↑𝐼))
2423oveq2d 7393 . . . . . . 7 (π‘₯ = 𝐼 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑𝐼)))
2524oveq1d 7392 . . . . . 6 (π‘₯ = 𝐼 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢))
2625mpteq2dv 5227 . . . . 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 2747 . . . 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 340 . . 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 46914 . . 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 482 . . . . . 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 46915 . . . . . 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 413 . . . 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 12622 . 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 407 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 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5208  β€˜cfv 6516  (class class class)co 7377  0cc0 11075  1c1 11076   + caddc 11078   Β· cmul 11080   βˆ’ cmin 11409  2c2 12232  β„•0cn0 12437  β†‘cexp 13992  IterCompcitco 46896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692  ax-inf2 9601  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-pss 3947  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-tr 5243  df-id 5551  df-eprel 5557  df-po 5565  df-so 5566  df-fr 5608  df-we 5610  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-pred 6273  df-ord 6340  df-on 6341  df-lim 6342  df-suc 6343  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-om 7823  df-2nd 7942  df-frecs 8232  df-wrecs 8263  df-recs 8337  df-rdg 8376  df-er 8670  df-en 8906  df-dom 8907  df-sdom 8908  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11411  df-neg 11412  df-nn 12178  df-2 12240  df-n0 12438  df-z 12524  df-uz 12788  df-seq 13932  df-exp 13993  df-itco 46898
This theorem is referenced by:  ackval3  46922
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