![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > iseqfcl | GIF version |
Description: Range of the recursive sequence builder. New proofs should use seqf 10026 instead (together with iseqsst 10031 or iseqseq3 10041 if need be). (Contributed by Jim Kingdon, 11-Apr-2022.) (New usage is discouraged.) |
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 2095 | . . 3 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝑀) | |
3 | fveq2 5340 | . . . . 5 ⊢ (𝑥 = 𝑀 → (𝐹‘𝑥) = (𝐹‘𝑀)) | |
4 | 3 | eleq1d 2163 | . . . 4 ⊢ (𝑥 = 𝑀 → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘𝑀) ∈ 𝑆)) |
5 | iseqfcl.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → (𝐹‘𝑥) ∈ 𝑆) | |
6 | 5 | ralrimiva 2458 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑍 (𝐹‘𝑥) ∈ 𝑆) |
7 | uzid 9132 | . . . . . 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 2188 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ 𝑍) |
11 | 4, 6, 10 | rspcdva 2741 | . . 3 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝑆) |
12 | 9 | eleq2i 2161 | . . . . 5 ⊢ (𝑥 ∈ 𝑍 ↔ 𝑥 ∈ (ℤ≥‘𝑀)) |
13 | 12, 5 | sylan2br 283 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆) |
14 | iseqfcl.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
15 | 13, 14 | iseqovex 10016 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝑆 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝑆) |
16 | eqid 2095 | . . 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 10017 | . . 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 9968 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):(ℤ≥‘𝑀)⟶𝑆) |
19 | 9 | feq2i 5189 | . 2 ⊢ (seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆 ↔ seq𝑀( + , 𝐹, 𝑆):(ℤ≥‘𝑀)⟶𝑆) |
20 | 18, 19 | sylibr 133 | 1 ⊢ (𝜑 → seq𝑀( + , 𝐹, 𝑆):𝑍⟶𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 〈cop 3469 ↦ cmpt 3921 ⟶wf 5045 ‘cfv 5049 (class class class)co 5690 ↦ cmpt2 5692 freccfrec 6193 1c1 7448 + caddc 7450 ℤcz 8848 ℤ≥cuz 9118 seqcseq4 10000 |
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 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-addcom 7542 ax-addass 7544 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-0id 7550 ax-rnegex 7551 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-ltadd 7558 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-inn 8521 df-n0 8772 df-z 8849 df-uz 9119 df-iseq 10002 |
This theorem is referenced by: iseqsst 10031 |
Copyright terms: Public domain | W3C validator |