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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 10060 | . 2 ⊢ seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) | |
2 | nfcv 2240 | . . . . . 6 ⊢ Ⅎ𝑥ℤ≥ | |
3 | nfseq.1 | . . . . . 6 ⊢ Ⅎ𝑥𝑀 | |
4 | 2, 3 | nffv 5363 | . . . . 5 ⊢ Ⅎ𝑥(ℤ≥‘𝑀) |
5 | nfcv 2240 | . . . . 5 ⊢ Ⅎ𝑥V | |
6 | nfcv 2240 | . . . . . 6 ⊢ Ⅎ𝑥(𝑧 + 1) | |
7 | nfcv 2240 | . . . . . . 7 ⊢ Ⅎ𝑥𝑤 | |
8 | nfseq.2 | . . . . . . 7 ⊢ Ⅎ𝑥 + | |
9 | nfseq.3 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐹 | |
10 | 9, 6 | nffv 5363 | . . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘(𝑧 + 1)) |
11 | 7, 8, 10 | nfov 5733 | . . . . . 6 ⊢ Ⅎ𝑥(𝑤 + (𝐹‘(𝑧 + 1))) |
12 | 6, 11 | nfop 3668 | . . . . 5 ⊢ Ⅎ𝑥〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉 |
13 | 4, 5, 12 | nfmpo 5772 | . . . 4 ⊢ Ⅎ𝑥(𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉) |
14 | 9, 3 | nffv 5363 | . . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑀) |
15 | 3, 14 | nfop 3668 | . . . 4 ⊢ Ⅎ𝑥〈𝑀, (𝐹‘𝑀)〉 |
16 | 13, 15 | nffrec 6223 | . . 3 ⊢ Ⅎ𝑥frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
17 | 16 | nfrn 4722 | . 2 ⊢ Ⅎ𝑥ran frec((𝑧 ∈ (ℤ≥‘𝑀), 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))〉), 〈𝑀, (𝐹‘𝑀)〉) |
18 | 1, 17 | nfcxfr 2237 | 1 ⊢ Ⅎ𝑥seq𝑀( + , 𝐹) |
Colors of variables: wff set class |
Syntax hints: Ⅎwnfc 2227 Vcvv 2641 〈cop 3477 ran crn 4478 ‘cfv 5059 (class class class)co 5706 ∈ cmpo 5708 freccfrec 6217 1c1 7501 + caddc 7503 ℤ≥cuz 9176 seqcseq 10059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-rab 2384 df-v 2643 df-un 3025 df-in 3027 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-xp 4483 df-cnv 4485 df-dm 4487 df-rn 4488 df-res 4489 df-iota 5024 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-recs 6132 df-frec 6218 df-seqfrec 10060 |
This theorem is referenced by: seq3f1olemstep 10115 seq3f1olemp 10116 nfsum1 10964 nfsum 10965 |
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