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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 17136 below, is useful for avoiding direct
reference to the hard-coded numeric index in component extractor
definitions, such as the 1 in df-base 17144 and the ;10 in
df-ple 17216, 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 17128 | . . 3 ⊢ (𝐸‘ndx) = 𝑁 |
4 | 3 | eqcomi 2733 | . 2 ⊢ 𝑁 = (𝐸‘ndx) |
5 | sloteq 17115 | . . 3 ⊢ (𝑁 = (𝐸‘ndx) → Slot 𝑁 = Slot (𝐸‘ndx)) | |
6 | 1, 5 | eqtrid 2776 | . 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 1533 ∈ wcel 2098 ‘cfv 6533 ℕcn 12209 Slot cslot 17113 ndxcnx 17125 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-1cn 11164 ax-addcl 11166 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-nn 12210 df-slot 17114 df-ndx 17126 |
This theorem is referenced by: strndxid 17130 setsidvaldOLD 17132 baseid 17146 2strop 17167 2strop1 17171 resslemOLD 17186 plusgid 17223 mulridx 17238 starvid 17247 scaid 17259 vscaid 17264 ipid 17275 tsetid 17297 pleid 17311 ocid 17326 dsid 17330 unifid 17340 homid 17356 ccoid 17358 oppglemOLD 19257 mgplemOLD 20034 opprlemOLD 20232 sralemOLD 21015 zlmlemOLD 21372 znbaslemOLD 21398 opsrbaslemOLD 21915 tnglemOLD 24472 itvid 28159 lngid 28160 ttglemOLD 28598 cchhllemOLD 28614 edgfid 28717 eufid 32857 resvlemOLD 32912 hlhilslemOLD 41300 mnringnmulrdOLD 43458 |
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