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Mirrors > Home > ILE Home > Th. List > ialgrp1 | GIF version |
Description: The value of the algorithm iterator 𝑅 at (𝐾 + 1). (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by Mario Carneiro, 27-Dec-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 |
---|---|
ialgrp1 | ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅‘𝐾))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | algrf.2 | . . . 4 ⊢ 𝑅 = seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆) | |
2 | 1 | fveq1i 5199 | . . 3 ⊢ (𝑅‘(𝐾 + 1)) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘(𝐾 + 1)) |
3 | simpr 108 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ 𝑍) | |
4 | algrf.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
5 | 3, 4 | syl6eleq 2171 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐾 ∈ (ℤ≥‘𝑀)) |
6 | algrf.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
7 | 6 | adantr 270 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝑆 ∈ 𝑉) |
8 | algrf.4 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
9 | 8 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐴 ∈ 𝑆) |
10 | 4, 9 | ialgrlemconst 10425 | . . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ 𝑍) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑍 × {𝐴})‘𝑥) ∈ 𝑆) |
11 | algrf.5 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑆⟶𝑆) | |
12 | 11 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → 𝐹:𝑆⟶𝑆) |
13 | 12 | ialgrlem1st 10424 | . . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ 𝑍) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐹 ∘ 1st )𝑦) ∈ 𝑆) |
14 | 5, 7, 10, 13 | iseqp1 9445 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
15 | 2, 14 | syl5eq 2125 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
16 | 5, 7, 10, 13 | iseqcl 9443 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾) ∈ 𝑆) |
17 | 4 | peano2uzs 8672 | . . . . . 6 ⊢ (𝐾 ∈ 𝑍 → (𝐾 + 1) ∈ 𝑍) |
18 | fvconst2g 5396 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐾 + 1) ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴) | |
19 | 8, 17, 18 | syl2an 283 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) = 𝐴) |
20 | 19, 9 | eqeltrd 2155 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) |
21 | algrflemg 5871 | . . . 4 ⊢ (((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾) ∈ 𝑆 ∧ ((𝑍 × {𝐴})‘(𝐾 + 1)) ∈ 𝑆) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾))) | |
22 | 16, 20, 21 | syl2anc 403 | . . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1))) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾))) |
23 | 1 | fveq1i 5199 | . . . 4 ⊢ (𝑅‘𝐾) = (seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾) |
24 | 23 | fveq2i 5201 | . . 3 ⊢ (𝐹‘(𝑅‘𝐾)) = (𝐹‘(seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)) |
25 | 22, 24 | syl6reqr 2132 | . 2 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝐹‘(𝑅‘𝐾)) = ((seq𝑀((𝐹 ∘ 1st ), (𝑍 × {𝐴}), 𝑆)‘𝐾)(𝐹 ∘ 1st )((𝑍 × {𝐴})‘(𝐾 + 1)))) |
26 | 15, 25 | eqtr4d 2116 | 1 ⊢ ((𝜑 ∧ 𝐾 ∈ 𝑍) → (𝑅‘(𝐾 + 1)) = (𝐹‘(𝑅‘𝐾))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 {csn 3398 × cxp 4361 ∘ ccom 4367 ⟶wf 4918 ‘cfv 4922 (class class class)co 5532 1st c1st 5785 1c1 6982 + caddc 6984 ℤcz 8351 ℤ≥cuz 8619 seqcseq 9431 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 df-uz 8620 df-iseq 9432 |
This theorem is referenced by: ialginv 10429 ialgcvg 10430 ialgcvga 10433 ialgfx 10434 |
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