Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 11930. (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 11910 | . 2 ⊢ 𝐸 = Slot (𝐸‘ndx) |
4 | 1, 2 | ndxarg 11909 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
5 | 4, 2 | eqeltri 2190 | . 2 ⊢ (𝐸‘ndx) ∈ ℕ |
6 | 3, 5 | pm3.2i 270 | 1 ⊢ (𝐸 = Slot (𝐸‘ndx) ∧ (𝐸‘ndx) ∈ ℕ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∈ wcel 1465 ‘cfv 5093 ℕcn 8688 ndxcnx 11883 Slot cslot 11885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-iota 5058 df-fun 5095 df-fv 5101 df-inn 8689 df-ndx 11889 df-slot 11890 |
This theorem is referenced by: base0 11935 baseslid 11942 plusgslid 11981 2stropg 11988 2strop1g 11991 mulrslid 11998 starvslid 12007 scaslid 12015 vscaslid 12018 ipslid 12026 tsetslid 12036 pleslid 12043 dsslid 12046 |
Copyright terms: Public domain | W3C validator |