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Theorem nfiseq 9747
 Description: Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Hypotheses
Ref Expression
nfiseq.1 𝑥𝑀
nfiseq.2 𝑥 +
nfiseq.3 𝑥𝐹
nfiseq.4 𝑥𝑆
Assertion
Ref Expression
nfiseq 𝑥seq𝑀( + , 𝐹, 𝑆)

Proof of Theorem nfiseq
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iseq 9741 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2223 . . . . . 6 𝑥
3 nfiseq.1 . . . . . 6 𝑥𝑀
42, 3nffv 5260 . . . . 5 𝑥(ℤ𝑀)
5 nfiseq.4 . . . . 5 𝑥𝑆
6 nfcv 2223 . . . . . 6 𝑥(𝑦 + 1)
7 nfcv 2223 . . . . . . 7 𝑥𝑧
8 nfiseq.2 . . . . . . 7 𝑥 +
9 nfiseq.3 . . . . . . . 8 𝑥𝐹
109, 6nffv 5260 . . . . . . 7 𝑥(𝐹‘(𝑦 + 1))
117, 8, 10nfov 5614 . . . . . 6 𝑥(𝑧 + (𝐹‘(𝑦 + 1)))
126, 11nfop 3612 . . . . 5 𝑥⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩
134, 5, 12nfmpt2 5652 . . . 4 𝑥(𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩)
149, 3nffv 5260 . . . . 5 𝑥(𝐹𝑀)
153, 14nfop 3612 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1613, 15nffrec 6093 . . 3 𝑥frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4638 . 2 𝑥ran frec((𝑦 ∈ (ℤ𝑀), 𝑧𝑆 ↦ ⟨(𝑦 + 1), (𝑧 + (𝐹‘(𝑦 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2220 1 𝑥seq𝑀( + , 𝐹, 𝑆)
 Colors of variables: wff set class Syntax hints:  Ⅎwnfc 2210  ⟨cop 3425  ran crn 4402  ‘cfv 4969  (class class class)co 5591   ↦ cmpt2 5593  freccfrec 6087  1c1 7254   + caddc 7256  ℤ≥cuz 8914  seqcseq 9740 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 2065 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-un 2988  df-in 2990  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-iota 4934  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-recs 6002  df-frec 6088  df-iseq 9741 This theorem is referenced by:  nfsum1  10567  nfsum  10568
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