Theorem ndxslid 13226

Description: A structure component extractor is defined by its own index. That the index is a natural number will also be needed in quite a few contexts so it is included in the conclusion of this theorem which can be used as a hypothesis of theorems like strslfv 13246. (Contributed by Jim Kingdon, 29-Jan-2023.)

Hypotheses

Ref	Expression
ndxarg.1	⊢ 𝐸 = Slot 𝑁
ndxarg.2	⊢ 𝑁 ∈ ℕ

Assertion

Ref	Expression
ndxslid	⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Proof of Theorem ndxslid

Step	Hyp	Ref	Expression
1		ndxarg.1	. . 3 ⊢ 𝐸 = Slot 𝑁
2		ndxarg.2	. . 3 ⊢ 𝑁 ∈ ℕ
3	1, 2	ndxid 13225	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 13224	. . 3 ⊢ (𝐸‘ndx) = 𝑁
5	4, 2	eqeltri 2305	. 2 ⊢ (𝐸‘ndx) ∈ ℕ
6	3, 5	pm3.2i 272	1 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ‘cfv 5351  ℕcn 9233  ndxcnx 13198  Slot cslot 13200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fv 5359  df-inn 9234  df-ndx 13204  df-slot 13205

This theorem is referenced by:  base0  13251  baseslid  13259  plusgslid  13314  2stropg  13323  2strop1g  13326  mulrslid  13334  starvslid  13343  scaslid  13355  vscaslid  13365  ipslid  13373  tsetslid  13390  pleslid  13404  dsslid  13419  homslid  13437  ccoslid  13440  prdsbaslemss  13476  zlmlemg  14763  znbaslemnn  14774  iedgvalg  15999  iedgex  16001  edgfiedgval2dom  16017  setsiedg  16034  iedgval0  16036  edgvalg  16041  edgstruct  16046