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Theorem nfseq 10255
 Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfseq.1 𝑥𝑀
nfseq.2 𝑥 +
nfseq.3 𝑥𝐹
Assertion
Ref Expression
nfseq 𝑥seq𝑀( + , 𝐹)

Proof of Theorem nfseq
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-seqfrec 10246 . 2 seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2282 . . . . . 6 𝑥
3 nfseq.1 . . . . . 6 𝑥𝑀
42, 3nffv 5435 . . . . 5 𝑥(ℤ𝑀)
5 nfcv 2282 . . . . 5 𝑥V
6 nfcv 2282 . . . . . 6 𝑥(𝑧 + 1)
7 nfcv 2282 . . . . . . 7 𝑥𝑤
8 nfseq.2 . . . . . . 7 𝑥 +
9 nfseq.3 . . . . . . . 8 𝑥𝐹
109, 6nffv 5435 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
117, 8, 10nfov 5805 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
126, 11nfop 3725 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
134, 5, 12nfmpo 5844 . . . 4 𝑥(𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
149, 3nffv 5435 . . . . 5 𝑥(𝐹𝑀)
153, 14nfop 3725 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1613, 15nffrec 6297 . . 3 𝑥frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4788 . 2 𝑥ran frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2279 1 𝑥seq𝑀( + , 𝐹)
 Colors of variables: wff set class Syntax hints:  Ⅎwnfc 2269  Vcvv 2687  ⟨cop 3531  ran crn 4544  ‘cfv 5127  (class class class)co 5778   ∈ cmpo 5780  freccfrec 6291  1c1 7641   + caddc 7643  ℤ≥cuz 9346  seqcseq 10245 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-un 3076  df-in 3078  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-mpt 3995  df-xp 4549  df-cnv 4551  df-dm 4553  df-rn 4554  df-res 4555  df-iota 5092  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-recs 6206  df-frec 6292  df-seqfrec 10246 This theorem is referenced by:  seq3f1olemstep  10301  seq3f1olemp  10302  nfsum1  11153  nfsum  11154  nfcprod1  11351  nfcprod  11352
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