Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itcovalt2 Structured version   Visualization version   GIF version

Theorem itcovalt2 47450
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 6890 . . . . 5 (π‘₯ = 0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜0))
2 oveq2 7419 . . . . . . . 8 (π‘₯ = 0 β†’ (2↑π‘₯) = (2↑0))
32oveq2d 7427 . . . . . . 7 (π‘₯ = 0 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑0)))
43oveq1d 7426 . . . . . 6 (π‘₯ = 0 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑0)) βˆ’ 𝐢))
54mpteq2dv 5249 . . . . 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 2746 . . . 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 339 . . 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 6890 . . . . 5 (π‘₯ = 𝑦 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜π‘¦))
9 oveq2 7419 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (2↑π‘₯) = (2↑𝑦))
109oveq2d 7427 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑𝑦)))
1110oveq1d 7426 . . . . . 6 (π‘₯ = 𝑦 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑𝑦)) βˆ’ 𝐢))
1211mpteq2dv 5249 . . . . 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 2746 . . . 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 339 . . 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 6890 . . . . 5 (π‘₯ = (𝑦 + 1) β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)))
16 oveq2 7419 . . . . . . . 8 (π‘₯ = (𝑦 + 1) β†’ (2↑π‘₯) = (2↑(𝑦 + 1)))
1716oveq2d 7427 . . . . . . 7 (π‘₯ = (𝑦 + 1) β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))))
1817oveq1d 7426 . . . . . 6 (π‘₯ = (𝑦 + 1) β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑(𝑦 + 1))) βˆ’ 𝐢))
1918mpteq2dv 5249 . . . . 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 2746 . . . 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 339 . . 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 6890 . . . . 5 (π‘₯ = 𝐼 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜πΌ))
23 oveq2 7419 . . . . . . . 8 (π‘₯ = 𝐼 β†’ (2↑π‘₯) = (2↑𝐼))
2423oveq2d 7427 . . . . . . 7 (π‘₯ = 𝐼 β†’ ((𝑛 + 𝐢) Β· (2↑π‘₯)) = ((𝑛 + 𝐢) Β· (2↑𝐼)))
2524oveq1d 7426 . . . . . 6 (π‘₯ = 𝐼 β†’ (((𝑛 + 𝐢) Β· (2↑π‘₯)) βˆ’ 𝐢) = (((𝑛 + 𝐢) Β· (2↑𝐼)) βˆ’ 𝐢))
2625mpteq2dv 5249 . . . . 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 2746 . . . 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 339 . . 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 47448 . . 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 480 . . . . . 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 47449 . . . . . 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 411 . . . 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 12661 . 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 405 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 394   = wceq 1539   ∈ wcel 2104   ↦ cmpt 5230  β€˜cfv 6542  (class class class)co 7411  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   βˆ’ cmin 11448  2c2 12271  β„•0cn0 12476  β†‘cexp 14031  IterCompcitco 47430
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-inf2 9638  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-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-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  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-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-seq 13971  df-exp 14032  df-itco 47432
This theorem is referenced by:  ackval3  47456
  Copyright terms: Public domain W3C validator