![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 5430 | . . 3 ⊢ (𝑅‘(𝐾 + 1)) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘(𝐾 + 1)) |
3 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ 𝑍) | |
4 | algrf.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | eleqtrdi 2233 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ (ℤ≥‘𝑀)) |
6 | algrf.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
7 | 6 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
8 | 4, 7 | ialgrlemconst 11760 | . . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ 𝑍) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
9 | algrf.5 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
10 | 9 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐹:𝑆⟶𝑆) |
11 | 10 | ialgrlem1st 11759 | . . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
12 | 5, 8, 11 | seq3p1 10266 | . . 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 | syl5eq 2185 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
14 | 1 | fveq1i 5430 | . . . 4 ⊢ (𝑅‘𝐾) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾) |
15 | 14 | oveq1i 5792 | . . 3 ⊢ ((𝑅‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) |
16 | algrf.3 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
17 | 4, 1, 16, 6, 9 | algrf 11762 | . . . . 5 ⊢ (𝜑 → 𝑅:𝑍⟶𝑆) |
18 | 17 | ffvelrnda 5563 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘𝐾) ∈ 𝑆) |
19 | 4 | peano2uzs 9406 | . . . . . 6 ⊢ (𝐾 ∈ 𝑍 → (𝐾 + 1) ∈ 𝑍) |
20 | fvconst2g 5642 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐾 + 1) ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴) | |
21 | 6, 19, 20 | syl2an 287 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴) |
22 | 21, 7 | eqeltrd 2217 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) |
23 | algrflemg 6135 | . . . 4 ⊢ (((𝑅‘𝐾) ∈ 𝑆 ∧ ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) → ((𝑅‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(𝑅‘𝐾))) | |
24 | 18, 22, 23 | syl2anc 409 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑅‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(𝑅‘𝐾))) |
25 | 15, 24 | syl5reqr 2188 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐹‘(𝑅‘𝐾)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
26 | 13, 25 | eqtr4d 2176 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅‘𝐾))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 {csn 3532 × cxp 4545 ∘ ccom 4551 ⟶wf 5127 ‘cfv 5131 (class class class)co 5782 1st c1st 6044 1c1 7645 + caddc 7647 ℤcz 9078 ℤ≥cuz 9350 seqcseq 10249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-uz 9351 df-seqfrec 10250 |
This theorem is referenced by: alginv 11764 algcvg 11765 algcvga 11768 algfx 11769 |
Copyright terms: Public domain | W3C validator |