Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 6756 | . . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0)) | |
3 | 2 | feq1d 6569 | . . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘0):𝐴⟶𝐴)) |
4 | fveq2 6756 | . . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦)) | |
5 | 4 | feq1d 6569 | . . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴)) |
6 | fveq2 6756 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1))) | |
7 | 6 | feq1d 6569 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘(𝑦 + 1)):𝐴⟶𝐴)) |
8 | fveq2 6756 | . . . 4 ⊢ (𝑥 = 𝑁 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑁)) | |
9 | 8 | feq1d 6569 | . . 3 ⊢ (𝑥 = 𝑁 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴)) |
10 | f1oi 6737 | . . . . . 6 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
11 | f1of 6700 | . . . . . 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 6595 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
15 | 14 | reseq2d 5880 | . . . . . 6 ⊢ (𝜑 → ( I ↾ dom 𝐹) = ( I ↾ 𝐴)) |
16 | 15 | feq1d 6569 | . . . . 5 ⊢ (𝜑 → (( I ↾ dom 𝐹):𝐴⟶𝐴 ↔ ( I ↾ 𝐴):𝐴⟶𝐴)) |
17 | 12, 16 | mpbird 256 | . . . 4 ⊢ (𝜑 → ( I ↾ dom 𝐹):𝐴⟶𝐴) |
18 | itcovalendof.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
19 | 13, 18 | fexd 7085 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V) |
20 | itcoval0 45896 | . . . . . 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 6569 | . . . 4 ⊢ (𝜑 → (((IterComp‘𝐹)‘0):𝐴⟶𝐴 ↔ ( I ↾ dom 𝐹):𝐴⟶𝐴)) |
23 | 17, 22 | mpbird 256 | . . 3 ⊢ (𝜑 → ((IterComp‘𝐹)‘0):𝐴⟶𝐴) |
24 | 13 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝐹:𝐴⟶𝐴) |
25 | simpr 484 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) | |
26 | 24, 25 | fcod 6610 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → (𝐹 ∘ ((IterComp‘𝐹)‘𝑦)):𝐴⟶𝐴) |
27 | 19 | ad2antrr 722 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝐹 ∈ V) |
28 | simplr 765 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝑦 ∈ ℕ0) | |
29 | eqidd 2739 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘𝑦) = ((IterComp‘𝐹)‘𝑦)) | |
30 | itcovalsucov 45902 | . . . . . 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 6569 | . . . 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 256 | . . 3 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘(𝑦 + 1)):𝐴⟶𝐴) |
34 | 3, 5, 7, 9, 23, 33 | nn0indd 12347 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
35 | 1, 34 | mpdan 683 | 1 ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 Vcvv 3422 I cid 5479 dom cdm 5580 ↾ cres 5582 ∘ ccom 5584 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 (class class class)co 7255 0cc0 10802 1c1 10803 + caddc 10805 ℕ0cn0 12163 IterCompcitco 45891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-seq 13650 df-itco 45893 |
This theorem is referenced by: ackendofnn0 45918 ackvalsucsucval 45922 |
Copyright terms: Public domain | W3C validator |