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Theorem nfseq 13977
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-seq 13968 . 2 seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
2 nfcv 2895 . . . . 5 𝑥V
3 nfcv 2895 . . . . . 6 𝑥(𝑧 + 1)
4 nfcv 2895 . . . . . . 7 𝑥𝑤
5 nfseq.2 . . . . . . 7 𝑥 +
6 nfseq.3 . . . . . . . 8 𝑥𝐹
76, 3nffv 6892 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
84, 5, 7nfov 7432 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
93, 8nfop 4882 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
102, 2, 9nfmpo 7484 . . . 4 𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11 nfseq.1 . . . . 5 𝑥𝑀
126, 11nffv 6892 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4882 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 8410 . . 3 𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2895 . . 3 𝑥ω
1614, 15nfima 6058 . 2 𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
171, 16nfcxfr 2893 1 𝑥seq𝑀( + , 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2875  Vcvv 3466  cop 4627  cima 5670  cfv 6534  (class class class)co 7402  cmpo 7404  ωcom 7849  reccrdg 8405  1c1 11108   + caddc 11110  seqcseq 13967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-xp 5673  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-iota 6486  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-seq 13968
This theorem is referenced by:  seqof2  14027  nfsum1  15638  nfsum  15639  nfcprod1  15856  nfcprod  15857  lgamgulm2  26908  binomcxplemdvbinom  43661  binomcxplemdvsum  43663  binomcxplemnotnn0  43664  fmuldfeqlem1  44843  fmuldfeq  44844  sumnnodd  44891  stoweidlem51  45312
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