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Mirrors > Home > ILE Home > Th. List > iseqcl | GIF version |
Description: Closure properties of the recursive sequence builder. (Contributed by Jim Kingdon, 1-Jun-2020.) |
Ref | Expression |
---|---|
iseqcl.1 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
iseqcl.ex | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
iseqcl.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
iseqcl.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
iseqcl | ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqcl.1 | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
2 | eluzel2 8624 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
4 | eqid 2081 | . . 3 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) | |
5 | iseqcl.ex | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
6 | uzid 8633 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
7 | 3, 6 | syl 14 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
8 | iseqcl.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) | |
9 | 8 | ralrimiva 2434 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆) |
10 | fveq2 5198 | . . . . . 6 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
11 | 10 | eleq1d 2147 | . . . . 5 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
12 | 11 | rspcv 2697 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆 → (𝐹‘𝑀) ∈ 𝑆)) |
13 | 7, 9, 12 | sylc 61 | . . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
14 | iseqcl.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
15 | 8, 14 | iseqovex 9439 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
16 | eqid 2081 | . . 3 ⊢ frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) | |
17 | 16, 8, 14 | iseqval 9440 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ 𝑆 ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉)) |
18 | 3, 4, 5, 13, 15, 16, 17 | frecuzrdgcl 9415 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆) |
19 | 1, 18 | mpdan 412 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹, 𝑆)‘𝑁) ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 ∀wral 2348 〈cop 3401 ↦ cmpt 3839 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 freccfrec 6000 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: iseqf 9444 iseqp1 9445 isermono 9457 iseqsplit 9458 iseqcaopr2 9461 iseqid3 9465 iseqhomo 9468 iseqz 9469 iseqdistr 9470 serige0 9473 serile 9474 expivallem 9477 expival 9478 ibcval5 9690 ialgrp1 10428 |
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