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| 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 | oveq2 7439 | . . . . 5 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s )) | |
| 2 | 1 | eleq1d 2826 | . . . 4 ⊢ (𝑚 = 0s → ((𝐴↑s𝑚) ∈ No ↔ (𝐴↑s 0s ) ∈ No )) | 
| 3 | 2 | imbi2d 340 | . . 3 ⊢ (𝑚 = 0s → ((𝐴 ∈ No → (𝐴↑s𝑚) ∈ No ) ↔ (𝐴 ∈ No → (𝐴↑s 0s ) ∈ No ))) | 
| 4 | oveq2 7439 | . . . . 5 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛)) | |
| 5 | 4 | eleq1d 2826 | . . . 4 ⊢ (𝑚 = 𝑛 → ((𝐴↑s𝑚) ∈ No ↔ (𝐴↑s𝑛) ∈ No )) | 
| 6 | 5 | imbi2d 340 | . . 3 ⊢ (𝑚 = 𝑛 → ((𝐴 ∈ No → (𝐴↑s𝑚) ∈ No ) ↔ (𝐴 ∈ No → (𝐴↑s𝑛) ∈ No ))) | 
| 7 | oveq2 7439 | . . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s ))) | |
| 8 | 7 | eleq1d 2826 | . . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴↑s𝑚) ∈ No ↔ (𝐴↑s(𝑛 +s 1s )) ∈ No )) | 
| 9 | 8 | imbi2d 340 | . . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 ∈ No → (𝐴↑s𝑚) ∈ No ) ↔ (𝐴 ∈ No → (𝐴↑s(𝑛 +s 1s )) ∈ No ))) | 
| 10 | oveq2 7439 | . . . . 5 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁)) | |
| 11 | 10 | eleq1d 2826 | . . . 4 ⊢ (𝑚 = 𝑁 → ((𝐴↑s𝑚) ∈ No ↔ (𝐴↑s𝑁) ∈ No )) | 
| 12 | 11 | imbi2d 340 | . . 3 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ No → (𝐴↑s𝑚) ∈ No ) ↔ (𝐴 ∈ No → (𝐴↑s𝑁) ∈ No ))) | 
| 13 | exps0 28410 | . . . 4 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s ) | |
| 14 | 1sno 27872 | . . . 4 ⊢ 1s ∈ No | |
| 15 | 13, 14 | eqeltrdi 2849 | . . 3 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) ∈ No ) | 
| 16 | simp2 1138 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ No ∧ (𝐴↑s𝑛) ∈ No ) → 𝐴 ∈ No ) | |
| 17 | simp1 1137 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ No ∧ (𝐴↑s𝑛) ∈ No ) → 𝑛 ∈ ℕ0s) | |
| 18 | expsp1 28412 | . . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴)) | |
| 19 | 16, 17, 18 | syl2anc 584 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ No ∧ (𝐴↑s𝑛) ∈ No ) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴)) | 
| 20 | simp3 1139 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ No ∧ (𝐴↑s𝑛) ∈ No ) → (𝐴↑s𝑛) ∈ No ) | |
| 21 | 20, 16 | mulscld 28161 | . . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ No ∧ (𝐴↑s𝑛) ∈ No ) → ((𝐴↑s𝑛) ·s 𝐴) ∈ No ) | 
| 22 | 19, 21 | eqeltrd 2841 | . . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ No ∧ (𝐴↑s𝑛) ∈ No ) → (𝐴↑s(𝑛 +s 1s )) ∈ No ) | 
| 23 | 22 | 3exp 1120 | . . . 4 ⊢ (𝑛 ∈ ℕ0s → (𝐴 ∈ No → ((𝐴↑s𝑛) ∈ No → (𝐴↑s(𝑛 +s 1s )) ∈ No ))) | 
| 24 | 23 | a2d 29 | . . 3 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ No → (𝐴↑s𝑛) ∈ No ) → (𝐴 ∈ No → (𝐴↑s(𝑛 +s 1s )) ∈ No ))) | 
| 25 | 3, 6, 9, 12, 15, 24 | n0sind 28337 | . 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ No → (𝐴↑s𝑁) ∈ No )) | 
| 26 | 25 | impcom 407 | 1 ⊢ ((𝐴 ∈ No ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ No ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 No csur 27684 0s c0s 27867 1s c1s 27868 +s cadds 27992 ·s cmuls 28132 ℕ0scnn0s 28318 ↑scexps 28396 | 
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| 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-rmo 3380 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-tp 4631 df-op 4633 df-ot 4635 df-uni 4908 df-int 4947 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-se 5638 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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-nadd 8704 df-no 27687 df-slt 27688 df-bday 27689 df-sle 27790 df-sslt 27826 df-scut 27828 df-0s 27869 df-1s 27870 df-made 27886 df-old 27887 df-left 27889 df-right 27890 df-norec 27971 df-norec2 27982 df-adds 27993 df-negs 28053 df-subs 28054 df-muls 28133 df-seqs 28290 df-n0s 28320 df-nns 28321 df-zs 28365 df-exps 28397 | 
| This theorem is referenced by: expsne0 28414 expsgt0 28415 cutpw2 28417 pw2bday 28418 pw2cut 28420 zs12bday 28424 | 
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