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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 17238 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 17246 and the ;10 in
df-ple 17318, 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 17230 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
4 | 3 | eqcomi 2744 | . 2 ⊢ 𝑁 = (𝐸‘ndx) |
5 | sloteq 17217 | . . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx)) | |
6 | 1, 5 | eqtrid 2787 | . 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 1537 ∈ wcel 2106 ‘cfv 6563 ℕcn 12264 Slot cslot 17215 ndxcnx 17227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 |
This theorem is referenced by: strndxid 17232 setsidvaldOLD 17234 baseid 17248 2strop 17269 2strop1 17273 resslemOLD 17288 plusgid 17325 mulridx 17340 starvid 17349 scaid 17361 vscaid 17366 ipid 17377 tsetid 17399 pleid 17413 ocid 17428 dsid 17432 unifid 17442 homid 17458 ccoid 17460 oppglemOLD 19382 mgplemOLD 20157 opprlemOLD 20357 sralemOLD 21194 zlmlemOLD 21546 znbaslemOLD 21572 opsrbaslemOLD 22086 tnglemOLD 24670 itvid 28462 lngid 28463 ttglemOLD 28901 cchhllemOLD 28917 edgfid 29020 eufid 33275 resvlemOLD 33338 hlhilslemOLD 41922 mnringnmulrdOLD 44206 |
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