| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ndxid | Structured version Visualization version GIF version | ||
| Description: A structure component
extractor is defined by its own index. This
theorem, together with strfv 17240 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 17248 and the ;10 in
df-ple 17317, making it easier to change should the need
arise.
For example, we can refer to a specific poset with base set 𝐵 and order relation 𝐿 using {〈(Base‘ndx), 𝐵〉, 〈(le‘ndx), 𝐿〉} rather than {〈1, 𝐵〉, 〈;10, 𝐿〉}. The latter, while shorter to state, requires revision if we later change ;10 to some other number, and it may also be harder to remember. (Contributed by NM, 19-Oct-2012.) (Revised by Mario Carneiro, 6-Oct-2013.) (Proof shortened by BJ, 27-Dec-2021.) |
| Ref | Expression |
|---|---|
| ndxarg.e | ⊢ 𝐸 = Slot 𝑁 |
| ndxarg.n | ⊢ 𝑁 ∈ ℕ |
| Ref | Expression |
|---|---|
| ndxid | ⊢ 𝐸 = Slot (𝐸‘ndx) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ndxarg.e | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
| 2 | ndxarg.n | . . . 4 ⊢ 𝑁 ∈ ℕ | |
| 3 | 1, 2 | ndxarg 17233 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
| 4 | 3 | eqcomi 2746 | . 2 ⊢ 𝑁 = (𝐸‘ndx) |
| 5 | sloteq 17220 | . . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx)) | |
| 6 | 1, 5 | eqtrid 2789 | . 2 ⊢ (𝑁 = (𝐸‘ndx) → 𝐸 = Slot (𝐸‘ndx)) |
| 7 | 4, 6 | ax-mp 5 | 1 ⊢ 𝐸 = Slot (𝐸‘ndx) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 ℕcn 12266 Slot cslot 17218 ndxcnx 17230 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 |
| This theorem is referenced by: strndxid 17235 baseid 17250 2strop 17269 2strop1 17273 resslemOLD 17288 plusgid 17324 mulridx 17338 starvid 17347 scaid 17359 vscaid 17364 ipid 17375 tsetid 17397 pleid 17411 ocid 17426 dsid 17430 unifid 17440 homid 17456 ccoid 17458 oppglemOLD 19369 opprlemOLD 20340 sralemOLD 21176 zlmlemOLD 21528 znbaslemOLD 21554 opsrbaslemOLD 22068 tnglemOLD 24654 itvid 28447 lngid 28448 ttglemOLD 28886 cchhllemOLD 28902 edgfid 29005 eufid 33294 resvlemOLD 33358 hlhilslemOLD 41941 mnringnmulrdOLD 44229 |
| Copyright terms: Public domain | W3C validator |