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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 Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
algrf.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
algrf.2 | ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆) |
algrf.3 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
algrf.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) |
algrf.5 | ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) |
algrf.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
Ref | Expression |
---|---|
ialgr0 | ⊢ (𝜑 → (𝑅‘𝑀) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrf.2 | . . 3 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆) | |
2 | 1 | fveq1i 5252 | . 2 ⊢ (𝑅‘𝑀) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘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 10803 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
7 | algrf.5 | . . . . 5 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
8 | 7 | ialgrlem1st 10802 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
9 | 3, 6, 8 | iseq1 9750 | . . 3 ⊢ (𝜑 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝑀) = ((𝑍 × {𝐴})‘𝑀)) |
10 | uzid 8926 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
11 | 3, 10 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
12 | 11, 4 | syl6eleqr 2176 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
13 | fvconst2g 5449 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑀 ∈ 𝑍) → ((𝑍 × {𝐴})‘𝑀) = 𝐴) | |
14 | 5, 12, 13 | syl2anc 403 | . . 3 ⊢ (𝜑 → ((𝑍 × {𝐴})‘𝑀) = 𝐴) |
15 | 9, 14 | eqtrd 2115 | . 2 ⊢ (𝜑 → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝑀) = 𝐴) |
16 | 2, 15 | syl5eq 2127 | 1 ⊢ (𝜑 → (𝑅‘𝑀) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 ∈ wcel 1434 {csn 3422 × cxp 4397 ∘ ccom 4403 ⟶wf 4963 ‘cfv 4967 1st c1st 5842 ℤcz 8644 ℤ≥cuz 8912 seqcseq 9738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-addcom 7346 ax-addass 7348 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-0id 7354 ax-rnegex 7355 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-ltadd 7362 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-inn 8315 df-n0 8564 df-z 8645 df-uz 8913 df-iseq 9739 |
This theorem is referenced by: ialgcvg 10808 eucialg 10819 |
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