Theorem ndxslid 12730

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 12750. (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 12729	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 12728	. . 3 ⊢ (𝐸‘ndx) = 𝑁
5	4, 2	eqeltri 2269	. 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 2167  ‘cfv 5259  ℕcn 9009  ndxcnx 12702  Slot cslot 12704

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7989  ax-resscn 7990  ax-1re 7992  ax-addrcl 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-inn 9010  df-ndx 12708  df-slot 12709

This theorem is referenced by:  base0  12755  baseslid  12762  plusgslid  12817  2stropg  12825  2strop1g  12828  mulrslid  12836  starvslid  12845  scaslid  12857  vscaslid  12867  ipslid  12875  tsetslid  12892  pleslid  12906  dsslid  12921  homslid  12939  ccoslid  12942  prdsbaslemss  12978  zlmlemg  14262  znbaslemnn  14273