Theorem ndxslid 13023

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 13043. (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 13022	. 2 ⊢ 𝐸 = Slot (𝐸‘ndx)
4	1, 2	ndxarg 13021	. . 3 ⊢ (𝐸‘ndx) = 𝑁
5	4, 2	eqeltri 2282	. 2 ⊢ (𝐸‘ndx) ∈ ℕ
6	3, 5	pm3.2i 272	1 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ)

Syntax hints:   ∧ wa 104   = wceq 1375   ∈ wcel 2180  ‘cfv 5294  ℕcn 9078  ndxcnx 12995  Slot cslot 12997

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1re 8061  ax-addrcl 8064

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-sbc 3009  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fv 5302  df-inn 9079  df-ndx 13001  df-slot 13002

This theorem is referenced by:  base0  13048  baseslid  13056  plusgslid  13111  2stropg  13120  2strop1g  13123  mulrslid  13131  starvslid  13140  scaslid  13152  vscaslid  13162  ipslid  13170  tsetslid  13187  pleslid  13201  dsslid  13216  homslid  13234  ccoslid  13237  prdsbaslemss  13273  zlmlemg  14557  znbaslemnn  14568  iedgvalg  15783  iedgex  15785  edgfiedgval2dom  15801  setsiedg  15818  iedgval0  15820  edgvalg  15825  edgstruct  15829