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Mirrors > Home > ILE Home > Th. List > seq1cd | GIF version |
Description: Initial value of the recursive sequence builder. A version of seq3-1 10455 which provides two classes 𝐷 and 𝐶 for the terms and the value being accumulated, respectively. (Contributed by Jim Kingdon, 19-Jul-2023.) |
Ref | Expression |
---|---|
seq1cd.1 | ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶) |
seq1cd.2 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶) |
seq1cd.4 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
seq1cd.5 | ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷) |
Ref | Expression |
---|---|
seq1cd | ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | seq1cd.4 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
2 | seq1cd.1 | . 2 ⊢ (𝜑 → (𝐹‘𝑀) ∈ 𝐶) | |
3 | ssv 3177 | . . 3 ⊢ 𝐶 ⊆ V | |
4 | 3 | a1i 9 | . 2 ⊢ (𝜑 → 𝐶 ⊆ V) |
5 | seq1cd.5 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝑥) ∈ 𝐷) | |
6 | seq1cd.2 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) → (𝑥 + 𝑦) ∈ 𝐶) | |
7 | 5, 6 | seqovcd 10458 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑦 ∈ 𝐶)) → (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦) ∈ 𝐶) |
8 | iseqvalcbv 10452 | . 2 ⊢ frec((𝑠 ∈ (ℤ≥‘𝑀), 𝑡 ∈ V ↦ 〈(𝑠 + 1), (𝑠(𝑢 ∈ (ℤ≥‘𝑀), 𝑣 ∈ 𝐶 ↦ (𝑣 + (𝐹‘(𝑢 + 1))))𝑡)〉), 〈𝑀, (𝐹‘𝑀)〉) = frec((𝑥 ∈ (ℤ≥‘𝑀), 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ 𝐶 ↦ (𝑤 + (𝐹‘(𝑧 + 1))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) | |
9 | 1, 8, 2, 6, 5 | seqvalcd 10454 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) = ran frec((𝑠 ∈ (ℤ≥‘𝑀), 𝑡 ∈ V ↦ 〈(𝑠 + 1), (𝑠(𝑢 ∈ (ℤ≥‘𝑀), 𝑣 ∈ 𝐶 ↦ (𝑣 + (𝐹‘(𝑢 + 1))))𝑡)〉), 〈𝑀, (𝐹‘𝑀)〉)) |
10 | 1, 2, 4, 7, 8, 9 | frecuzrdg0t 10417 | 1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 ⊆ wss 3129 〈cop 3595 ‘cfv 5215 (class class class)co 5872 ∈ cmpo 5874 freccfrec 6388 1c1 7809 + caddc 7811 ℤcz 9249 ℤ≥cuz 9524 seqcseq 10440 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-ltadd 7924 |
This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-recs 6303 df-frec 6389 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-inn 8916 df-n0 9173 df-z 9250 df-uz 9525 df-seqfrec 10441 |
This theorem is referenced by: ennnfonelem0 12398 |
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