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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 11785. (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 11765 | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 11764 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
5 | 4, 2 | eqeltri 2172 | . 2 ⊢ (𝐸‘ndx) ∈ ℕ |
6 | 3, 5 | pm3.2i 268 | 1 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1299 ∈ wcel 1448 ‘cfv 5059 ℕcn 8578 ndxcnx 11738 Slot cslot 11740 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-cnex 7586 ax-resscn 7587 ax-1re 7589 ax-addrcl 7592 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-iota 5024 df-fun 5061 df-fv 5067 df-inn 8579 df-ndx 11744 df-slot 11745 |
This theorem is referenced by: base0 11790 baseslid 11797 plusgslid 11836 2stropg 11843 2strop1g 11846 mulrslid 11853 starvslid 11862 scaslid 11870 vscaslid 11873 ipslid 11881 tsetslid 11891 pleslid 11898 dsslid 11901 |
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