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Mirrors > Home > ILE Home > Th. List > ndxslid | GIF version |
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.) |
Ref | Expression |
---|---|
ndxarg.1 | ⊢ 𝐸 = Slot 𝑁 |
ndxarg.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
ndxslid | ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
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) |
Colors of variables: wff set class |
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 |
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