ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nfseq GIF version

Theorem nfseq 10069
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 10060 . 2 seq𝑀( + , 𝐹) = ran frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2240 . . . . . 6 𝑥
3 nfseq.1 . . . . . 6 𝑥𝑀
42, 3nffv 5363 . . . . 5 𝑥(ℤ𝑀)
5 nfcv 2240 . . . . 5 𝑥V
6 nfcv 2240 . . . . . 6 𝑥(𝑧 + 1)
7 nfcv 2240 . . . . . . 7 𝑥𝑤
8 nfseq.2 . . . . . . 7 𝑥 +
9 nfseq.3 . . . . . . . 8 𝑥𝐹
109, 6nffv 5363 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
117, 8, 10nfov 5733 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
126, 11nfop 3668 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
134, 5, 12nfmpo 5772 . . . 4 𝑥(𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
149, 3nffv 5363 . . . . 5 𝑥(𝐹𝑀)
153, 14nfop 3668 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1613, 15nffrec 6223 . . 3 𝑥frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4722 . 2 𝑥ran frec((𝑧 ∈ (ℤ𝑀), 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2237 1 𝑥seq𝑀( + , 𝐹)
Colors of variables: wff set class
Syntax hints:  wnfc 2227  Vcvv 2641  cop 3477  ran crn 4478  cfv 5059  (class class class)co 5706  cmpo 5708  freccfrec 6217  1c1 7501   + caddc 7503  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
  Copyright terms: Public domain W3C validator