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Mirrors > Home > ILE Home > Th. List > iseqfcl | GIF version |
Description: Range of the recursive sequence builder. (Contributed by Jim Kingdon, 11-Apr-2022.) |
Ref | Expression |
---|---|
iseqfcl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
iseqfcl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iseqfcl.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) |
iseqfcl.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
Ref | Expression |
---|---|
iseqfcl | ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseqfcl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | eqid 2083 | . . 3 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) | |
3 | fveq2 5252 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
4 | 3 | eleq1d 2151 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
5 | iseqfcl.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) | |
6 | 5 | ralrimiva 2440 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 (𝐹‘𝑥) ∈ 𝑆) |
7 | uzid 8927 | . . . . . 6 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
8 | 1, 7 | syl 14 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
9 | iseqfcl.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
10 | 8, 9 | syl6eleqr 2176 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
11 | 4, 6, 10 | rspcdva 2717 | . . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
12 | 9 | eleq2i 2149 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↔ 𝑥 ∈ (ℤ≥‘𝑀)) |
13 | 12, 5 | sylan2br 282 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
14 | iseqfcl.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
15 | 13, 14 | iseqovex 9747 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
16 | eqid 2083 | . . 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, 13, 14 | iseqval 9748 | . . 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 | 1, 2, 11, 15, 16, 17 | frecuzrdgtcl 9707 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):(ℤ≥‘𝑀)⟶𝑆) |
19 | 9 | feq2i 5107 | . 2 ⊢ (seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆 ↔ seq𝑀( + , 𝐹, 𝑆):(ℤ≥‘𝑀)⟶𝑆) |
20 | 18, 19 | sylibr 132 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∈ wcel 1434 〈cop 3425 ↦ cmpt 3865 ⟶wf 4964 ‘cfv 4968 (class class class)co 5590 ↦ cmpt2 5592 freccfrec 6086 1c1 7253 + caddc 7255 ℤcz 8645 ℤ≥cuz 8913 seqcseq 9739 |
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 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-addcom 7347 ax-addass 7349 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-0id 7355 ax-rnegex 7356 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-ltadd 7363 |
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 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-f1 4973 df-fo 4974 df-f1o 4975 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-recs 6001 df-frec 6087 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-inn 8316 df-n0 8565 df-z 8646 df-uz 8914 df-iseq 9740 |
This theorem is referenced by: iseqoveq 9758 iseqss 9759 iseqsst 9760 iseqfeq2 9763 iseqfeq 9765 iserf 9767 iserfre 9768 facnn 9969 fac0 9970 resqrexlemf 10266 ialgrf 10806 |
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