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Mirrors > Home > ILE Home > Th. List > nfiseq | GIF version |
Description: Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.) |
Ref | Expression |
---|---|
nfiseq.1 | ⊢ Ⅎ𝑥𝑀 |
nfiseq.2 | ⊢ Ⅎ𝑥 + |
nfiseq.3 | ⊢ Ⅎ𝑥𝐹 |
nfiseq.4 | ⊢ Ⅎ𝑥𝑆 |
Ref | Expression |
---|---|
nfiseq | ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iseq 9522 | . 2 ⊢ seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ 〈(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | nfcv 2220 | . . . . . 6 ⊢ Ⅎ𝑥ℤ≥ | |
3 | nfiseq.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑀 | |
4 | 2, 3 | nffv 5216 | . . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀) |
5 | nfiseq.4 | . . . . 5 ⊢ Ⅎ𝑥𝑆 | |
6 | nfcv 2220 | . . . . . 6 ⊢ Ⅎ𝑥(𝑦 + 1) | |
7 | nfcv 2220 | . . . . . . 7 ⊢ Ⅎ𝑥𝑧 | |
8 | nfiseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
9 | nfiseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
10 | 9, 6 | nffv 5216 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑦 + 1)) |
11 | 7, 8, 10 | nfov 5566 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + (𝐹‘(𝑦 + 1))) |
12 | 6, 11 | nfop 3594 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))〉 |
13 | 4, 5, 12 | nfmpt2 5604 | . . . 4 ⊢ Ⅎ𝑥(𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ 〈(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))〉) |
14 | 9, 3 | nffv 5216 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
15 | 3, 14 | nfop 3594 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
16 | 13, 15 | nffrec 6045 | . . 3 ⊢ Ⅎ𝑥frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ 〈(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
17 | 16 | nfrn 4607 | . 2 ⊢ Ⅎ𝑥ran frec((𝑦 ∈ (ℤ≥‘𝑀), 𝑧 ∈ 𝑆 ↦ 〈(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
18 | 1, 17 | nfcxfr 2217 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹, 𝑆) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2207 〈cop 3409 ran crn 4372 ‘cfv 4932 (class class class)co 5543 ↦ cmpt2 5545 freccfrec 6039 1c1 7044 + caddc 7046 ℤ≥cuz 8700 seqcseq 9521 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1687 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ral 2354 df-rex 2355 df-rab 2358 df-v 2604 df-un 2978 df-in 2980 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-br 3794 df-opab 3848 df-mpt 3849 df-xp 4377 df-cnv 4379 df-dm 4381 df-rn 4382 df-res 4383 df-iota 4897 df-fv 4940 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-recs 5954 df-frec 6040 df-iseq 9522 |
This theorem is referenced by: nfsum1 10331 nfsum 10332 |
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