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Theorem nfseq 13972
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 13963 . 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 6891 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
84, 5, 7nfov 7431 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
93, 8nfop 4881 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
102, 2, 9nfmpo 7483 . . . 4 𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11 nfseq.1 . . . . 5 𝑥𝑀
126, 11nffv 6891 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4881 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 8409 . . 3 𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2895 . . 3 𝑥ω
1614, 15nfima 6057 . 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 4626  cima 5669  cfv 6533  (class class class)co 7401  cmpo 7403  ωcom 7848  reccrdg 8404  1c1 11106   + caddc 11108  seqcseq 13962
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 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-xp 5672  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-iota 6485  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-seq 13963
This theorem is referenced by:  seqof2  14022  nfsum1  15632  nfsum  15633  nfcprod1  15850  nfcprod  15851  lgamgulm2  26883  binomcxplemdvbinom  43567  binomcxplemdvsum  43569  binomcxplemnotnn0  43570  fmuldfeqlem1  44749  fmuldfeq  44750  sumnnodd  44797  stoweidlem51  45218
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