Theorem ndxslid 13100

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 13120. (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 13099	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 13098	. . 3 ⊢ (𝐸‘ndx) = 𝑁
5	4, 2	eqeltri 2302	. 2 ⊢ (𝐸‘ndx) ∈ ℕ
6	3, 5	pm3.2i 272	1 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ‘cfv 5324  ℕcn 9136  ndxcnx 13072  Slot cslot 13074

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8116  ax-resscn 8117  ax-1re 8119  ax-addrcl 8122

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-inn 9137  df-ndx 13078  df-slot 13079

This theorem is referenced by:  base0  13125  baseslid  13133  plusgslid  13188  2stropg  13197  2strop1g  13200  mulrslid  13208  starvslid  13217  scaslid  13229  vscaslid  13239  ipslid  13247  tsetslid  13264  pleslid  13278  dsslid  13293  homslid  13311  ccoslid  13314  prdsbaslemss  13350  zlmlemg  14635  znbaslemnn  14646  iedgvalg  15861  iedgex  15863  edgfiedgval2dom  15879  setsiedg  15896  iedgval0  15898  edgvalg  15903  edgstruct  15908