| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > expscl | Structured version Visualization version GIF version | ||
| Description: Closure law for surreal exponentiation. (Contributed by Scott Fenton, 7-Aug-2025.) |
| Ref | Expression |
|---|---|
| expscl | ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ssid 3971 | . 2 ⊢ No ⊆ No | |
| 2 | mulscl 28043 | . 2 ⊢ ((𝑥 ∈ No ∧ 𝑦 ∈ No ) → (𝑥 ·s 𝑦) ∈ No ) | |
| 3 | 1sno 27745 | . 2 ⊢ 1s ∈ No | |
| 4 | 1, 2, 3 | expscllem 28322 | 1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 (class class class)co 7389 No csur 27557 ℕ0scnn0s 28212 ↑scexps 28304 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4913 df-iun 4959 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-om 7845 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-oadd 8440 df-nadd 8632 df-no 27560 df-slt 27561 df-bday 27562 df-sle 27663 df-sslt 27699 df-scut 27701 df-0s 27742 df-1s 27743 df-made 27761 df-old 27762 df-left 27764 df-right 27765 df-norec 27851 df-norec2 27862 df-adds 27873 df-negs 27933 df-subs 27934 df-muls 28016 df-seqs 28184 df-n0s 28214 df-nns 28215 df-zs 28273 df-exps 28305 |
| This theorem is referenced by: expadds 28326 expsne0 28327 expsgt0 28328 pw2recs 28329 pw2divscld 28330 pw2divsmuld 28331 pw2divscan3d 28332 pw2divscan2d 28333 pw2gt0divsd 28334 pw2ge0divsd 28335 pw2divsrecd 28336 pw2divsnegd 28338 pw2cut 28341 zs12ge0 28348 zs12bday 28349 |
| Copyright terms: Public domain | W3C validator |