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Mirrors > Home > ILE Home > Th. List > nfseq | GIF version |
Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfseq.1 | ⊢ Ⅎ𝑥𝑀 |
nfseq.2 | ⊢ Ⅎ𝑥 + |
nfseq.3 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
nfseq | ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-seqfrec 10485 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑥ℤ≥ | |
3 | nfseq.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑀 | |
4 | 2, 3 | nffv 5547 | . . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀) |
5 | nfcv 2332 | . . . . 5 ⊢ Ⅎ𝑥V | |
6 | nfcv 2332 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
7 | nfcv 2332 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
8 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
9 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
10 | 9, 6 | nffv 5547 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
11 | 7, 8, 10 | nfov 5930 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
12 | 6, 11 | nfop 3812 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
13 | 4, 5, 12 | nfmpo 5969 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
14 | 9, 3 | nffv 5547 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
15 | 3, 14 | nfop 3812 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
16 | 13, 15 | nffrec 6425 | . . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
17 | 16 | nfrn 4893 | . 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
18 | 1, 17 | nfcxfr 2329 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2319 Vcvv 2752 〈cop 3613 ran crn 4648 ‘cfv 5238 (class class class)co 5900 ∈ cmpo 5902 freccfrec 6419 1c1 7847 + caddc 7849 ℤ≥cuz 9563 seqcseq 10484 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-un 3148 df-in 3150 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-mpt 4084 df-xp 4653 df-cnv 4655 df-dm 4657 df-rn 4658 df-res 4659 df-iota 5199 df-fv 5246 df-ov 5903 df-oprab 5904 df-mpo 5905 df-recs 6334 df-frec 6420 df-seqfrec 10485 |
This theorem is referenced by: seq3f1olemstep 10540 seq3f1olemp 10541 nfsum1 11405 nfsum 11406 nfcprod1 11603 nfcprod 11604 |
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