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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > itcovalsucov | Structured version Visualization version GIF version |
Description: The value of the function that returns the n-th iterate of a function with regard to composition at a successor. (Contributed by AV, 4-May-2024.) |
Ref | Expression |
---|---|
itcovalsucov | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹 ∘ 𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itcovalsuc 45431 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹)) | |
2 | eqidd 2760 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔)) = (𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))) | |
3 | coeq2 5691 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺)) | |
4 | 3 | ad2antrl 728 | . . 3 ⊢ (((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) ∧ (𝑔 = 𝐺 ∧ 𝑗 = 𝐹)) → (𝐹 ∘ 𝑔) = (𝐹 ∘ 𝐺)) |
5 | id 22 | . . . . . 6 ⊢ (𝐺 = ((IterComp‘𝐹)‘𝑌) → 𝐺 = ((IterComp‘𝐹)‘𝑌)) | |
6 | fvex 6664 | . . . . . 6 ⊢ ((IterComp‘𝐹)‘𝑌) ∈ V | |
7 | 5, 6 | eqeltrdi 2859 | . . . . 5 ⊢ (𝐺 = ((IterComp‘𝐹)‘𝑌) → 𝐺 ∈ V) |
8 | 7 | eqcoms 2767 | . . . 4 ⊢ (((IterComp‘𝐹)‘𝑌) = 𝐺 → 𝐺 ∈ V) |
9 | 8 | 3ad2ant3 1133 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → 𝐺 ∈ V) |
10 | elex 3427 | . . . 4 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V) | |
11 | 10 | 3ad2ant1 1131 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → 𝐹 ∈ V) |
12 | 8 | anim2i 620 | . . . . 5 ⊢ ((𝐹 ∈ 𝑉 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ V)) |
13 | 12 | 3adant2 1129 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ V)) |
14 | coexg 7632 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ V) → (𝐹 ∘ 𝐺) ∈ V) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → (𝐹 ∘ 𝐺) ∈ V) |
16 | 2, 4, 9, 11, 15 | ovmpod 7290 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → (𝐺(𝑔 ∈ V, 𝑗 ∈ V ↦ (𝐹 ∘ 𝑔))𝐹) = (𝐹 ∘ 𝐺)) |
17 | 1, 16 | eqtrd 2794 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑌 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑌) = 𝐺) → ((IterComp‘𝐹)‘(𝑌 + 1)) = (𝐹 ∘ 𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1085 = wceq 1539 ∈ wcel 2112 Vcvv 3407 ∘ ccom 5521 ‘cfv 6328 (class class class)co 7143 ∈ cmpo 7145 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: itcovalendof 45433 itcovalpclem2 45435 itcovalt2lem2 45440 ackvalsucsucval 45452 |
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