Theorem zz12s 28843

Description: A surreal integer is a dyadic fraction. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
zz12s	⊢ (𝐴 ∈ ℤs → 𝐴 ∈ ℤs[1/2])

Proof of Theorem zz12s

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2no 28787	. . . . . 6 ⊢ 2s ∈ No
2		exps0 28795	. . . . . 6 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
3	1, 2	ax-mp 5	. . . . 5 ⊢ (2s↑s 0s ) = 1s
4	3	oveq2i 7423	. . . 4 ⊢ (𝐴 /su (2s↑s 0s )) = (𝐴 /su 1s )
5		zno 28750	. . . . 5 ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
6	5	divs1d 28573	. . . 4 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = 𝐴)
7	4, 6	eqtr2id 2809	. . 3 ⊢ (𝐴 ∈ ℤs → 𝐴 = (𝐴 /su (2s↑s 0s )))
8		0n0s 28697	. . . 4 ⊢ 0s ∈ ℕ0s
9		oveq1 7419	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 /su (2s↑s𝑦)) = (𝐴 /su (2s↑s𝑦)))
10	9	eqeq2d 2772	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝐴 = (𝑥 /su (2s↑s𝑦)) ↔ 𝐴 = (𝐴 /su (2s↑s𝑦))))
11		oveq2 7420	. . . . . . 7 ⊢ (𝑦 = 0s → (2s↑s𝑦) = (2s↑s 0s ))
12	11	oveq2d 7428	. . . . . 6 ⊢ (𝑦 = 0s → (𝐴 /su (2s↑s𝑦)) = (𝐴 /su (2s↑s 0s )))
13	12	eqeq2d 2772	. . . . 5 ⊢ (𝑦 = 0s → (𝐴 = (𝐴 /su (2s↑s𝑦)) ↔ 𝐴 = (𝐴 /su (2s↑s 0s ))))
14	10, 13	rspc2ev 3589	. . . 4 ⊢ ((𝐴 ∈ ℤs ∧ 0s ∈ ℕ0s ∧ 𝐴 = (𝐴 /su (2s↑s 0s ))) → ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
15	8, 14	mp3an2 1478	. . 3 ⊢ ((𝐴 ∈ ℤs ∧ 𝐴 = (𝐴 /su (2s↑s 0s ))) → ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
16	7, 15	mpdan 700	. 2 ⊢ (𝐴 ∈ ℤs → ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
17		elz12s 28840	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
18	16, 17	sylibr 237	1 ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ ℤs[1/2])

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  (class class class)co 7412   No csur 27979   0_s c0s 28173   1_s c1s 28174   /_su cdivs 28555  ℕ_0scn0s 28680  ℤ_sczs 28746  2_sc2s 28778  ↑_scexps 28780  ℤ_s[1/2]cz12s 28782

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-nadd 8659  df-no 27982  df-lts 27983  df-bday 27984  df-les 28084  df-slts 28126  df-cuts 28128  df-0s 28175  df-1s 28176  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec 28306  df-norec2 28317  df-adds 28328  df-negs 28389  df-subs 28390  df-muls 28475  df-divs 28556  df-seqs 28652  df-n0s 28682  df-nns 28683  df-zs 28747  df-2s 28779  df-exps 28781  df-z12s 28783

This theorem is used by:  bdayfinlem  28854