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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  itcovalpc Structured version   Visualization version   GIF version
Theorem itcovalpc 47512
Description: The value of the function that returns the n-th iterate of the "plus a constant" function with regard to composition. (Contributed by AV, 4-May-2024.)
Hypothesis
Ref Expression
itcovalpc.f 𝐹 = (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢))
Assertion
Ref Expression
itcovalpc ((𝐼 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝐼))))
Distinct variable groups:   𝐢,𝑛   𝑛,𝐼
Allowed substitution hint:   𝐹(𝑛)
Proof of Theorem itcovalpc
Dummy variables 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6881 . . . 4 (π‘₯ = 0 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜0))
2 oveq2 7409 . . . . . 6 (π‘₯ = 0 β†’ (𝐢 Β· π‘₯) = (𝐢 Β· 0))
32oveq2d 7417 . . . . 5 (π‘₯ = 0 β†’ (𝑛 + (𝐢 Β· π‘₯)) = (𝑛 + (𝐢 Β· 0)))
43mpteq2dv 5240 . . . 4 (π‘₯ = 0 β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 0))))
51, 4eqeq12d 2740 . . 3 (π‘₯ = 0 β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) ↔ ((IterCompβ€˜πΉ)β€˜0) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 0)))))
6 fveq2 6881 . . . 4 (π‘₯ = 𝑦 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜π‘¦))
7 oveq2 7409 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝐢 Β· π‘₯) = (𝐢 Β· 𝑦))
87oveq2d 7417 . . . . 5 (π‘₯ = 𝑦 β†’ (𝑛 + (𝐢 Β· π‘₯)) = (𝑛 + (𝐢 Β· 𝑦)))
98mpteq2dv 5240 . . . 4 (π‘₯ = 𝑦 β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))))
106, 9eqeq12d 2740 . . 3 (π‘₯ = 𝑦 β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) ↔ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))))
11 fveq2 6881 . . . 4 (π‘₯ = (𝑦 + 1) β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)))
12 oveq2 7409 . . . . . 6 (π‘₯ = (𝑦 + 1) β†’ (𝐢 Β· π‘₯) = (𝐢 Β· (𝑦 + 1)))
1312oveq2d 7417 . . . . 5 (π‘₯ = (𝑦 + 1) β†’ (𝑛 + (𝐢 Β· π‘₯)) = (𝑛 + (𝐢 Β· (𝑦 + 1))))
1413mpteq2dv 5240 . . . 4 (π‘₯ = (𝑦 + 1) β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1)))))
1511, 14eqeq12d 2740 . . 3 (π‘₯ = (𝑦 + 1) β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) ↔ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1))))))
16 fveq2 6881 . . . 4 (π‘₯ = 𝐼 β†’ ((IterCompβ€˜πΉ)β€˜π‘₯) = ((IterCompβ€˜πΉ)β€˜πΌ))
17 oveq2 7409 . . . . . 6 (π‘₯ = 𝐼 β†’ (𝐢 Β· π‘₯) = (𝐢 Β· 𝐼))
1817oveq2d 7417 . . . . 5 (π‘₯ = 𝐼 β†’ (𝑛 + (𝐢 Β· π‘₯)) = (𝑛 + (𝐢 Β· 𝐼)))
1918mpteq2dv 5240 . . . 4 (π‘₯ = 𝐼 β†’ (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝐼))))
2016, 19eqeq12d 2740 . . 3 (π‘₯ = 𝐼 β†’ (((IterCompβ€˜πΉ)β€˜π‘₯) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· π‘₯))) ↔ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝐼)))))
21 itcovalpc.f . . . 4 𝐹 = (𝑛 ∈ β„•0 ↦ (𝑛 + 𝐢))
2221itcovalpclem1 47510 . . 3 (𝐢 ∈ β„•0 β†’ ((IterCompβ€˜πΉ)β€˜0) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 0))))
2321itcovalpclem2 47511 . . . . 5 ((𝑦 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ (((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1))))))
2423ancoms 458 . . . 4 ((𝐢 ∈ β„•0 ∧ 𝑦 ∈ β„•0) β†’ (((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1))))))
2524imp 406 . . 3 (((𝐢 ∈ β„•0 ∧ 𝑦 ∈ β„•0) ∧ ((IterCompβ€˜πΉ)β€˜π‘¦) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝑦)))) β†’ ((IterCompβ€˜πΉ)β€˜(𝑦 + 1)) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· (𝑦 + 1)))))
265, 10, 15, 20, 22, 25nn0indd 12655 . 2 ((𝐢 ∈ β„•0 ∧ 𝐼 ∈ β„•0) β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝐼))))
2726ancoms 458 1 ((𝐼 ∈ β„•0 ∧ 𝐢 ∈ β„•0) β†’ ((IterCompβ€˜πΉ)β€˜πΌ) = (𝑛 ∈ β„•0 ↦ (𝑛 + (𝐢 Β· 𝐼))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ↦ cmpt 5221  β€˜cfv 6533  (class class class)co 7401  0cc0 11105  1c1 11106   + caddc 11108   Β· cmul 11110  β„•0cn0 12468  IterCompcitco 47497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-seq 13963  df-itco 47499
This theorem is referenced by:  ackval1  47521  ackval2  47522
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