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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 6645 | . . . 4 ⊢ (𝑥 = 0 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘0)) | |
3 | 2 | feq1d 6472 | . . 3 ⊢ (𝑥 = 0 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘0):𝐴⟶𝐴)) |
4 | fveq2 6645 | . . . 4 ⊢ (𝑥 = 𝑦 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑦)) | |
5 | 4 | feq1d 6472 | . . 3 ⊢ (𝑥 = 𝑦 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴)) |
6 | fveq2 6645 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘(𝑦 + 1))) | |
7 | 6 | feq1d 6472 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘(𝑦 + 1)):𝐴⟶𝐴)) |
8 | fveq2 6645 | . . . 4 ⊢ (𝑥 = 𝑁 → ((IterComp‘𝐹)‘𝑥) = ((IterComp‘𝐹)‘𝑁)) | |
9 | 8 | feq1d 6472 | . . 3 ⊢ (𝑥 = 𝑁 → (((IterComp‘𝐹)‘𝑥):𝐴⟶𝐴 ↔ ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴)) |
10 | f1oi 6627 | . . . . . 6 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
11 | f1of 6590 | . . . . . 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 6497 | . . . . . . 7 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
15 | 14 | reseq2d 5818 | . . . . . 6 ⊢ (𝜑 → ( I ↾ dom 𝐹) = ( I ↾ 𝐴)) |
16 | 15 | feq1d 6472 | . . . . 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 6967 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ V) |
20 | itcoval0 45076 | . . . . . 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 6472 | . . . 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 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝐹:𝐴⟶𝐴) |
25 | simpr 488 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) | |
26 | 24, 25 | fcod 6506 | . . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → (𝐹 ∘ ((IterComp‘𝐹)‘𝑦)):𝐴⟶𝐴) |
27 | 19 | ad2antrr 725 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝐹 ∈ V) |
28 | simplr 768 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → 𝑦 ∈ ℕ0) | |
29 | eqidd 2799 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘𝑦) = ((IterComp‘𝐹)‘𝑦)) | |
30 | itcovalsucov 45082 | . . . . . 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 1368 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦):𝐴⟶𝐴) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ ((IterComp‘𝐹)‘𝑦))) |
32 | 31 | feq1d 6472 | . . . 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 12067 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
35 | 1, 34 | mpdan 686 | 1 ⊢ (𝜑 → ((IterComp‘𝐹)‘𝑁):𝐴⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 I cid 5424 dom cdm 5519 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 ℕ0cn0 11885 IterCompcitco 45071 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-itco 45073 |
This theorem is referenced by: ackendofnn0 45098 ackvalsucsucval 45102 |
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