![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ndxarg | Structured version Visualization version GIF version |
Description: Get the numeric argument from a defined structure component extractor such as df-base 15910. (Contributed by Mario Carneiro, 6-Oct-2013.) |
Ref | Expression |
---|---|
ndxarg.1 | ⊢ 𝐸 = Slot 𝑁 |
ndxarg.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
ndxarg | ⊢ (𝐸‘ndx) = 𝑁 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ndx 15907 | . . . 4 ⊢ ndx = ( I ↾ ℕ) | |
2 | nnex 11064 | . . . . 5 ⊢ ℕ ∈ V | |
3 | resiexg 7144 | . . . . 5 ⊢ (ℕ ∈ V → ( I ↾ ℕ) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ↾ ℕ) ∈ V |
5 | 1, 4 | eqeltri 2726 | . . 3 ⊢ ndx ∈ V |
6 | ndxarg.1 | . . 3 ⊢ 𝐸 = Slot 𝑁 | |
7 | 5, 6 | strfvn 15926 | . 2 ⊢ (𝐸‘ndx) = (ndx‘𝑁) |
8 | 1 | fveq1i 6230 | . 2 ⊢ (ndx‘𝑁) = (( I ↾ ℕ)‘𝑁) |
9 | ndxarg.2 | . . 3 ⊢ 𝑁 ∈ ℕ | |
10 | fvresi 6480 | . . 3 ⊢ (𝑁 ∈ ℕ → (( I ↾ ℕ)‘𝑁) = 𝑁) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ (( I ↾ ℕ)‘𝑁) = 𝑁 |
12 | 7, 8, 11 | 3eqtri 2677 | 1 ⊢ (𝐸‘ndx) = 𝑁 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 Vcvv 3231 I cid 5052 ↾ cres 5145 ‘cfv 5926 ℕcn 11058 ndxcnx 15901 Slot cslot 15903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-i2m1 10042 ax-1ne0 10043 ax-rrecex 10046 ax-cnre 10047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-nn 11059 df-ndx 15907 df-slot 15908 |
This theorem is referenced by: ndxid 15930 ndxidOLD 15931 basendx 15970 basendxnn 15971 resslem 15980 plusgndx 16023 2strstr 16030 2strstr1 16033 2strop1 16035 basendxnplusgndx 16036 mulrndx 16043 basendxnmulrndx 16046 starvndx 16051 scandx 16060 vscandx 16062 ipndx 16069 tsetndx 16087 plendx 16094 plendxOLD 16095 ocndx 16107 dsndx 16109 unifndx 16111 homndx 16121 ccondx 16123 slotsbhcdif 16127 oppglem 17826 mgplem 18540 opprlem 18674 rmodislmod 18979 sralem 19225 opsrbaslem 19525 opsrbaslemOLD 19526 zlmlem 19913 znbaslem 19934 znbaslemOLD 19935 tnglem 22491 itvndx 25384 lngndx 25385 ttglem 25801 cchhllem 25812 edgfndxnn 25915 baseltedgf 25917 resvlem 29959 hlhilslem 37547 |
Copyright terms: Public domain | W3C validator |