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 10371 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | nfcv 2306 | . . . . . 6 ⊢ Ⅎ𝑥ℤ≥ | |
3 | nfseq.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑀 | |
4 | 2, 3 | nffv 5490 | . . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀) |
5 | nfcv 2306 | . . . . 5 ⊢ Ⅎ𝑥V | |
6 | nfcv 2306 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
7 | nfcv 2306 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
8 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
9 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
10 | 9, 6 | nffv 5490 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
11 | 7, 8, 10 | nfov 5863 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
12 | 6, 11 | nfop 3768 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
13 | 4, 5, 12 | nfmpo 5902 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
14 | 9, 3 | nffv 5490 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
15 | 3, 14 | nfop 3768 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
16 | 13, 15 | nffrec 6355 | . . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
17 | 16 | nfrn 4843 | . 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
18 | 1, 17 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2293 Vcvv 2721 〈cop 3573 ran crn 4599 ‘cfv 5182 (class class class)co 5836 ∈ cmpo 5838 freccfrec 6349 1c1 7745 + caddc 7747 ℤ≥cuz 9457 seqcseq 10370 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-xp 4604 df-cnv 4606 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-seqfrec 10371 |
This theorem is referenced by: seq3f1olemstep 10426 seq3f1olemp 10427 nfsum1 11283 nfsum 11284 nfcprod1 11481 nfcprod 11482 |
Copyright terms: Public domain | W3C validator |