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Mirrors > Home > ILE Home > Th. List > ialgr0 | GIF version |
Description: The value of the algorithm iterator 𝑅 at 0 is the initial state 𝐴. (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 |
---|---|
ialgr0 | ⊢ (𝜑 → (𝑅‘𝑀) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrf.2 | . . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴})) | |
2 | 1 | fveq1i 5321 | . 2 ⊢ (𝑅‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) |
3 | algrf.3 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | algrf.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | algrf.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
6 | 4, 5 | ialgrlemconst 11366 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
7 | algrf.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
8 | 7 | ialgrlem1st 11365 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
9 | 3, 6, 8 | seq3-1 9940 | . . 3 ⊢ (𝜑 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = ((𝑍 × {𝐴})‘𝑀)) |
10 | uzid 9096 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
11 | 3, 10 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
12 | 11, 4 | syl6eleqr 2182 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
13 | fvconst2g 5527 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑀 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴) | |
14 | 5, 12, 13 | syl2anc 404 | . . 3 ⊢ (𝜑 → ((𝑍 × {𝐴})‘𝑀) = 𝐴) |
15 | 9, 14 | eqtrd 2121 | . 2 ⊢ (𝜑 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}))‘𝑀) = 𝐴) |
16 | 2, 15 | syl5eq 2133 | 1 ⊢ (𝜑 → (𝑅‘𝑀) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 {csn 3452 × cxp 4452 ∘ ccom 4458 ⟶wf 5026 ‘cfv 5030 1st c1st 5925 ℤcz 8813 ℤ≥cuz 9082 seqcseq 9915 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-addcom 7508 ax-addass 7510 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-0id 7516 ax-rnegex 7517 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-ltadd 7524 |
This theorem depends on definitions: df-bi 116 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-frec 6172 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-inn 8486 df-n0 8737 df-z 8814 df-uz 9083 df-iseq 9916 df-seq3 9917 |
This theorem is referenced by: algcvg 11371 eucalg 11382 |
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