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

Theorem nfiseq 8898
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 8893 . 2 seq𝑀( + , 𝐹, 𝑆) = ran frec((y (ℤ𝑀), z 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
2 nfcv 2175 . . . . . 6 x
3 nfiseq.1 . . . . . 6 x𝑀
42, 3nffv 5128 . . . . 5 x(ℤ𝑀)
5 nfiseq.4 . . . . 5 x𝑆
6 nfcv 2175 . . . . . 6 x(y + 1)
7 nfcv 2175 . . . . . . 7 xz
8 nfiseq.2 . . . . . . 7 x +
9 nfiseq.3 . . . . . . . 8 x𝐹
109, 6nffv 5128 . . . . . . 7 x(𝐹‘(y + 1))
117, 8, 10nfov 5478 . . . . . 6 x(z + (𝐹‘(y + 1)))
126, 11nfop 3556 . . . . 5 x⟨(y + 1), (z + (𝐹‘(y + 1)))⟩
134, 5, 12nfmpt2 5515 . . . 4 x(y (ℤ𝑀), z 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩)
149, 3nffv 5128 . . . . 5 x(𝐹𝑀)
153, 14nfop 3556 . . . 4 x𝑀, (𝐹𝑀)⟩
1613, 15nffrec 5921 . . 3 xfrec((y (ℤ𝑀), z 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
1716nfrn 4522 . 2 xran frec((y (ℤ𝑀), z 𝑆 ↦ ⟨(y + 1), (z + (𝐹‘(y + 1)))⟩), ⟨𝑀, (𝐹𝑀)⟩)
181, 17nfcxfr 2172 1 xseq𝑀( + , 𝐹, 𝑆)
Colors of variables: wff set class
Syntax hints:  wnfc 2162  cop 3370  ran crn 4289  cfv 4845  (class class class)co 5455  cmpt2 5457  freccfrec 5917  1c1 6712   + caddc 6714  cuz 8249  seqcseq 8892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-recs 5861  df-frec 5918  df-iseq 8893
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator