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Mirrors > Home > ILE Home > Th. List > algrp1 | GIF version |
Description: The value of the algorithm iterator 𝑅 at (𝐾 + 1). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Jim Kingdon, 12-Mar-2023.) |
Ref | Expression |
---|---|
algrf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
algrf.2 | ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) |
algrf.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
algrf.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
algrf.5 | ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) |
Ref | Expression |
---|---|
algrp1 | ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrf.2 | . . . 4 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) | |
2 | 1 | fveq1i 5512 | . . 3 ⊢ (𝑅‘(𝐾 + 1)) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝐾 + 1)) |
3 | simpr 110 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ 𝑍) | |
4 | algrf.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrdi 2270 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ (ℤ≥‘𝑀)) |
6 | algrf.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
7 | 6 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
8 | 4, 7 | ialgrlemconst 12026 | . . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ 𝑍) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
9 | algrf.5 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
10 | 9 | adantr 276 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐹:𝑆⟶𝑆) |
11 | 10 | ialgrlem1st 12025 | . . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
12 | 5, 8, 11 | seq3p1 10448 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
13 | 2, 12 | eqtrid 2222 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
14 | algrf.3 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
15 | 4, 1, 14, 6, 9 | algrf 12028 | . . . . 5 ⊢ (𝜑 → 𝑅:𝑍⟶𝑆) |
16 | 15 | ffvelcdmda 5647 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘𝐾) ∈ 𝑆) |
17 | 4 | peano2uzs 9573 | . . . . . 6 ⊢ (𝐾 ∈ 𝑍 → (𝐾 + 1) ∈ 𝑍) |
18 | fvconst2g 5726 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐾 + 1) ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴) | |
19 | 6, 17, 18 | syl2an 289 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴) |
20 | 19, 7 | eqeltrd 2254 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) |
21 | algrflemg 6225 | . . . 4 ⊢ (((𝑅‘𝐾) ∈ 𝑆 ∧ ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) → ((𝑅‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(𝑅‘𝐾))) | |
22 | 16, 20, 21 | syl2anc 411 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑅‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(𝑅‘𝐾))) |
23 | 1 | fveq1i 5512 | . . . 4 ⊢ (𝑅‘𝐾) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾) |
24 | 23 | oveq1i 5879 | . . 3 ⊢ ((𝑅‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) |
25 | 22, 24 | eqtr3di 2225 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐹‘(𝑅‘𝐾)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
26 | 13, 25 | eqtr4d 2213 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅‘𝐾))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {csn 3591 × cxp 4621 ∘ ccom 4627 ⟶wf 5208 ‘cfv 5212 (class class class)co 5869 1st c1st 6133 1c1 7803 + caddc 7805 ℤcz 9242 ℤ≥cuz 9517 seqcseq 10431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-n0 9166 df-z 9243 df-uz 9518 df-seqfrec 10432 |
This theorem is referenced by: alginv 12030 algcvg 12031 algcvga 12034 algfx 12035 |
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