Theorem ndxslid 12540

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 12560. (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 12539	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 12538	. . 3 ⊢ (𝐸‘ndx) = 𝑁
5	4, 2	eqeltri 2262	. 2 ⊢ (𝐸‘ndx) ∈ ℕ
6	3, 5	pm3.2i 272	1 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ‘cfv 5235  ℕcn 8950  ndxcnx 12512  Slot cslot 12514

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7933  ax-resscn 7934  ax-1re 7936  ax-addrcl 7939

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-inn 8951  df-ndx 12518  df-slot 12519

This theorem is referenced by:  base0  12565  baseslid  12572  plusgslid  12627  2stropg  12635  2strop1g  12638  mulrslid  12646  starvslid  12655  scaslid  12667  vscaslid  12677  ipslid  12685  tsetslid  12702  pleslid  12716  dsslid  12727  homslid  12744  ccoslid  12746  zlmlemg  13941  znbaslemnn  13952