Theorem expscllem 28843

Description: Lemma for proving non-negative surreal integer exponentiation closure. (Contributed by Scott Fenton, 7-Nov-2025.)

Hypotheses

Ref	Expression
expscllem.1	⊢ 𝐹 ⊆ No
expscllem.2	⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ·s 𝑦) ∈ 𝐹)
expscllem.3	⊢ 1s ∈ 𝐹

Assertion

Ref	Expression
expscllem	⊢ ((𝐴 ∈ 𝐹 ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ 𝐹)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑁,𝑦   𝑥,𝐹,𝑦

Proof of Theorem expscllem

Dummy variables 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7448	. . . . 5 ⊢ (𝑚 = 0s → (𝐴↑s𝑚) = (𝐴↑s 0s ))
2	1	eleq1d 2849	. . . 4 ⊢ (𝑚 = 0s → ((𝐴↑s𝑚) ∈ 𝐹 ↔ (𝐴↑s 0s ) ∈ 𝐹))
3	2	imbi2d 343	. . 3 ⊢ (𝑚 = 0s → ((𝐴 ∈ 𝐹 → (𝐴↑s𝑚) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑s 0s ) ∈ 𝐹)))
4		oveq2 7448	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐴↑s𝑚) = (𝐴↑s𝑛))
5	4	eleq1d 2849	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐴↑s𝑚) ∈ 𝐹 ↔ (𝐴↑s𝑛) ∈ 𝐹))
6	5	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐴 ∈ 𝐹 → (𝐴↑s𝑚) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑s𝑛) ∈ 𝐹)))
7		oveq2 7448	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴↑s𝑚) = (𝐴↑s(𝑛 +s 1s )))
8	7	eleq1d 2849	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴↑s𝑚) ∈ 𝐹 ↔ (𝐴↑s(𝑛 +s 1s )) ∈ 𝐹))
9	8	imbi2d 343	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 ∈ 𝐹 → (𝐴↑s𝑚) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑s(𝑛 +s 1s )) ∈ 𝐹)))
10		oveq2 7448	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐴↑s𝑚) = (𝐴↑s𝑁))
11	10	eleq1d 2849	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝐴↑s𝑚) ∈ 𝐹 ↔ (𝐴↑s𝑁) ∈ 𝐹))
12	11	imbi2d 343	. . 3 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ 𝐹 → (𝐴↑s𝑚) ∈ 𝐹) ↔ (𝐴 ∈ 𝐹 → (𝐴↑s𝑁) ∈ 𝐹)))
13		expscllem.1	. . . . . 6 ⊢ 𝐹 ⊆ No
14	13	sseli 3909	. . . . 5 ⊢ (𝐴 ∈ 𝐹 → 𝐴 ∈ No)
15		exps0 28840	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴↑s 0s ) = 1s )
16	14, 15	syl 18	. . . 4 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑s 0s ) = 1s )
17		expscllem.3	. . . 4 ⊢ 1s ∈ 𝐹
18	16, 17	eqeltrdi 2872	. . 3 ⊢ (𝐴 ∈ 𝐹 → (𝐴↑s 0s ) ∈ 𝐹)
19	14	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ 𝐹 ∧ (𝐴↑s𝑛) ∈ 𝐹) → 𝐴 ∈ No)
20		simp1 1154	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ 𝐹 ∧ (𝐴↑s𝑛) ∈ 𝐹) → 𝑛 ∈ ℕ0s)
21		expsp1 28842	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ 𝑛 ∈ ℕ0s) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
22	19, 20, 21	syl2anc 596	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ 𝐹 ∧ (𝐴↑s𝑛) ∈ 𝐹) → (𝐴↑s(𝑛 +s 1s )) = ((𝐴↑s𝑛) ·s 𝐴))
23		expscllem.2	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 ·s 𝑦) ∈ 𝐹)
24	23	caovcl 7636	. . . . . . . 8 ⊢ (((𝐴↑s𝑛) ∈ 𝐹 ∧ 𝐴 ∈ 𝐹) → ((𝐴↑s𝑛) ·s 𝐴) ∈ 𝐹)
25	24	ancoms 464	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐹 ∧ (𝐴↑s𝑛) ∈ 𝐹) → ((𝐴↑s𝑛) ·s 𝐴) ∈ 𝐹)
26	25	3adant1 1148	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ 𝐹 ∧ (𝐴↑s𝑛) ∈ 𝐹) → ((𝐴↑s𝑛) ·s 𝐴) ∈ 𝐹)
27	22, 26	eqeltrd 2864	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ 𝐹 ∧ (𝐴↑s𝑛) ∈ 𝐹) → (𝐴↑s(𝑛 +s 1s )) ∈ 𝐹)
28	27	3exp 1137	. . . 4 ⊢ (𝑛 ∈ ℕ0s → (𝐴 ∈ 𝐹 → ((𝐴↑s𝑛) ∈ 𝐹 → (𝐴↑s(𝑛 +s 1s )) ∈ 𝐹)))
29	28	a2d 30	. . 3 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ 𝐹 → (𝐴↑s𝑛) ∈ 𝐹) → (𝐴 ∈ 𝐹 → (𝐴↑s(𝑛 +s 1s )) ∈ 𝐹)))
30	3, 6, 9, 12, 18, 29	n0sind 28746	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ 𝐹 → (𝐴↑s𝑁) ∈ 𝐹))
31	30	impcom 413	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝑁 ∈ ℕ0s) → (𝐴↑s𝑁) ∈ 𝐹)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  (class class class)co 7440  Nocsur 28024   0_s c0s 28218   1_s c1s 28219   +_s cadds 28372   ·_s cmuls 28519  ℕ_0scn0s 28725  ↑_scexps 28825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-2o 8502  df-oadd 8505  df-nadd 8700  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-0s 28220  df-1s 28221  df-made 28240  df-old 28241  df-left 28243  df-right 28244  df-norec 28351  df-norec2 28362  df-adds 28373  df-negs 28434  df-subs 28435  df-muls 28520  df-seqs 28697  df-n0s 28727  df-nns 28728  df-zs 28792  df-exps 28826

This theorem is used by:  expscl  28844  n0expscl  28845  nnexpscl  28846  zexpscl  28847