Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 10219 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | nfcv 2281 | . . . . . 6 ⊢ Ⅎ𝑥ℤ≥ | |
3 | nfseq.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑀 | |
4 | 2, 3 | nffv 5431 | . . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀) |
5 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑥V | |
6 | nfcv 2281 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
7 | nfcv 2281 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
8 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
9 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
10 | 9, 6 | nffv 5431 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
11 | 7, 8, 10 | nfov 5801 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
12 | 6, 11 | nfop 3721 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
13 | 4, 5, 12 | nfmpo 5840 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
14 | 9, 3 | nffv 5431 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
15 | 3, 14 | nfop 3721 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
16 | 13, 15 | nffrec 6293 | . . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
17 | 16 | nfrn 4784 | . 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
18 | 1, 17 | nfcxfr 2278 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2268 Vcvv 2686 〈cop 3530 ran crn 4540 ‘cfv 5123 (class class class)co 5774 ∈ cmpo 5776 freccfrec 6287 1c1 7621 + caddc 7623 ℤ≥cuz 9326 seqcseq 10218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-un 3075 df-in 3077 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-xp 4545 df-cnv 4547 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-recs 6202 df-frec 6288 df-seqfrec 10219 |
This theorem is referenced by: seq3f1olemstep 10274 seq3f1olemp 10275 nfsum1 11125 nfsum 11126 nfcprod1 11323 nfcprod 11324 |
Copyright terms: Public domain | W3C validator |