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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > itcovalendof | Structured version Visualization version GIF version |
Description: The n-th iterate of an endofunction is an endofunction. (Contributed by AV, 7-May-2024.) |
Ref | Expression |
---|---|
itcovalendof.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
itcovalendof.f | ⊢ (𝜑 → 𝐹:𝐴⟶𝐴) |
itcovalendof.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
itcovalendof | ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itcovalendof.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | fveq2 6651 | . . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0)) | |
3 | 2 | feq1d 6476 | . . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘0):𝐴⟶𝐴)) |
4 | fveq2 6651 | . . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦)) | |
5 | 4 | feq1d 6476 | . . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴)) |
6 | fveq2 6651 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1))) | |
7 | 6 | feq1d 6476 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘(𝑦 + 1)):𝐴⟶𝐴)) |
8 | fveq2 6651 | . . . 4 ⊢ (𝑥 = 𝑁 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑁)) | |
9 | 8 | feq1d 6476 | . . 3 ⊢ (𝑥 = 𝑁 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴)) |
10 | f1oi 6632 | . . . . . 6 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
11 | f1of 6595 | . . . . . 6 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 → ( I ↾ 𝐴):𝐴⟶𝐴) | |
12 | 10, 11 | mp1i 13 | . . . . 5 ⊢ (𝜑 → ( I ↾ 𝐴):𝐴⟶𝐴) |
13 | itcovalendof.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶𝐴) | |
14 | 13 | fdmd 6501 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
15 | 14 | reseq2d 5816 | . . . . . 6 ⊢ (𝜑 → ( I ↾ dom 𝐹) = ( I ↾ 𝐴)) |
16 | 15 | feq1d 6476 | . . . . 5 ⊢ (𝜑 → (( I ↾ dom 𝐹):𝐴⟶𝐴 ↔ ( I ↾ 𝐴):𝐴⟶𝐴)) |
17 | 12, 16 | mpbird 260 | . . . 4 ⊢ (𝜑 → ( I ↾ dom 𝐹):𝐴⟶𝐴) |
18 | itcovalendof.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
19 | 13, 18 | fexd 6974 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V) |
20 | itcoval0 45426 | . . . . . 6 ⊢ (𝐹 ∈ V → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹)) | |
21 | 19, 20 | syl 17 | . . . . 5 ⊢ (𝜑 → ((IterComp‘𝐹)‘0) = ( I ↾ dom 𝐹)) |
22 | 21 | feq1d 6476 | . . . 4 ⊢ (𝜑 → (((IterComp‘𝐹)‘0):𝐴⟶𝐴 ↔ ( I ↾ dom 𝐹):𝐴⟶𝐴)) |
23 | 17, 22 | mpbird 260 | . . 3 ⊢ (𝜑 → ((IterComp‘𝐹)‘0):𝐴⟶𝐴) |
24 | 13 | ad2antrr 726 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝐹:𝐴⟶𝐴) |
25 | simpr 489 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) | |
26 | 24, 25 | fcod 6510 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → (𝐹 ∘ ((IterComp‘𝐹)‘𝑦)):𝐴⟶𝐴) |
27 | 19 | ad2antrr 726 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝐹 ∈ V) |
28 | simplr 769 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝑦 ∈ ℕ0) | |
29 | eqidd 2760 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘𝑦) = ((IterComp‘𝐹)‘𝑦)) | |
30 | itcovalsucov 45432 | . . . . . 6 ⊢ ((𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑦) = ((IterComp‘𝐹)‘𝑦)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ ((IterComp‘𝐹)‘𝑦))) | |
31 | 27, 28, 29, 30 | syl3anc 1369 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ ((IterComp‘𝐹)‘𝑦))) |
32 | 31 | feq1d 6476 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → (((IterComp‘𝐹)‘(𝑦 + 1)):𝐴⟶𝐴 ↔ (𝐹 ∘ ((IterComp‘𝐹)‘𝑦)):𝐴⟶𝐴)) |
33 | 26, 32 | mpbird 260 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘(𝑦 + 1)):𝐴⟶𝐴) |
34 | 3, 5, 7, 9, 23, 33 | nn0indd 12103 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
35 | 1, 34 | mpdan 687 | 1 ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 Vcvv 3407 I cid 5422 dom cdm 5517 ↾ cres 5519 ∘ ccom 5521 ⟶wf 6324 –1-1-onto→wf1o 6327 ‘cfv 6328 (class class class)co 7143 0cc0 10560 1c1 10561 + caddc 10563 ℕ0cn0 11919 IterCompcitco 45421 |
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 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 ax-inf2 9122 ax-cnex 10616 ax-resscn 10617 ax-1cn 10618 ax-icn 10619 ax-addcl 10620 ax-addrcl 10621 ax-mulcl 10622 ax-mulrcl 10623 ax-mulcom 10624 ax-addass 10625 ax-mulass 10626 ax-distr 10627 ax-i2m1 10628 ax-1ne0 10629 ax-1rid 10630 ax-rnegex 10631 ax-rrecex 10632 ax-cnre 10633 ax-pre-lttri 10634 ax-pre-lttrn 10635 ax-pre-ltadd 10636 ax-pre-mulgt0 10637 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-nel 3054 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-pss 3873 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-tp 4520 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-tr 5132 df-id 5423 df-eprel 5428 df-po 5436 df-so 5437 df-fr 5476 df-we 5478 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-pred 6119 df-ord 6165 df-on 6166 df-lim 6167 df-suc 6168 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-riota 7101 df-ov 7146 df-oprab 7147 df-mpo 7148 df-om 7573 df-2nd 7687 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8521 df-dom 8522 df-sdom 8523 df-pnf 10700 df-mnf 10701 df-xr 10702 df-ltxr 10703 df-le 10704 df-sub 10895 df-neg 10896 df-nn 11660 df-n0 11920 df-z 12006 df-uz 12268 df-seq 13404 df-itco 45423 |
This theorem is referenced by: ackendofnn0 45448 ackvalsucsucval 45452 |
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