MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfseq Structured version   Visualization version   GIF version

Theorem nfseq 13925
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 13916 . 2 seq𝑀( + , 𝐹) = (rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
2 nfcv 2904 . . . . 5 𝑥V
3 nfcv 2904 . . . . . 6 𝑥(𝑧 + 1)
4 nfcv 2904 . . . . . . 7 𝑥𝑤
5 nfseq.2 . . . . . . 7 𝑥 +
6 nfseq.3 . . . . . . . 8 𝑥𝐹
76, 3nffv 6856 . . . . . . 7 𝑥(𝐹‘(𝑧 + 1))
84, 5, 7nfov 7391 . . . . . 6 𝑥(𝑤 + (𝐹‘(𝑧 + 1)))
93, 8nfop 4850 . . . . 5 𝑥⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩
102, 2, 9nfmpo 7443 . . . 4 𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩)
11 nfseq.1 . . . . 5 𝑥𝑀
126, 11nffv 6856 . . . . 5 𝑥(𝐹𝑀)
1311, 12nfop 4850 . . . 4 𝑥𝑀, (𝐹𝑀)⟩
1410, 13nfrdg 8364 . . 3 𝑥rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
15 nfcv 2904 . . 3 𝑥ω
1614, 15nfima 6025 . 2 𝑥(rec((𝑧 ∈ V, 𝑤 ∈ V ↦ ⟨(𝑧 + 1), (𝑤 + (𝐹‘(𝑧 + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩) “ ω)
171, 16nfcxfr 2902 1 𝑥seq𝑀( + , 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2884  Vcvv 3447  cop 4596  cima 5640  cfv 6500  (class class class)co 7361  cmpo 7363  ωcom 7806  reccrdg 8359  1c1 11060   + caddc 11062  seqcseq 13915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-xp 5643  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-iota 6452  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seq 13916
This theorem is referenced by:  seqof2  13975  nfsum1  15583  nfsum  15584  nfcprod1  15801  nfcprod  15802  lgamgulm2  26408  binomcxplemdvbinom  42725  binomcxplemdvsum  42727  binomcxplemnotnn0  42728  fmuldfeqlem1  43913  fmuldfeq  43914  sumnnodd  43961  stoweidlem51  44382
  Copyright terms: Public domain W3C validator