Theorem zz12s 28649

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 28593	. . . . . 6 ⊢ 2s ∈ No
2		exps0 28601	. . . . . 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 28556	. . . . 5 ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
6	5	divs1d 28379	. . . 4 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = 𝐴)
7	4, 6	eqtr2id 2811	. . 3 ⊢ (𝐴 ∈ ℤs → 𝐴 = (𝐴 /su (2s↑s 0s )))
8		0n0s 28503	. . . 4 ⊢ 0s ∈ ℕ0s
9		oveq1 7419	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 /su (2s↑s𝑦)) = (𝐴 /su (2s↑s𝑦)))
10	9	eqeq2d 2774	. . . . 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 2774	. . . . 5 ⊢ (𝑦 = 0s → (𝐴 = (𝐴 /su (2s↑s𝑦)) ↔ 𝐴 = (𝐴 /su (2s↑s 0s ))))
14	10, 13	rspc2ev 3595	. . . 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 699	. 2 ⊢ (𝐴 ∈ ℤs → ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
17		elz12s 28646	. 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])

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  (class class class)co 7412   No csur 27785   0_s c0s 27979   1_s c1s 27980   /_su cdivs 28361  ℕ_0scn0s 28486  ℤ_sczs 28552  2_sc2s 28584  ↑_scexps 28586  ℤ_s[1/2]cz12s 28588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27788  df-lts 27789  df-bday 27790  df-les 27890  df-slts 27932  df-cuts 27934  df-0s 27981  df-1s 27982  df-made 28001  df-old 28002  df-left 28004  df-right 28005  df-norec 28112  df-norec2 28123  df-adds 28134  df-negs 28195  df-subs 28196  df-muls 28281  df-divs 28362  df-seqs 28458  df-n0s 28488  df-nns 28489  df-zs 28553  df-2s 28585  df-exps 28587  df-z12s 28589

This theorem is referenced by:  bdayfinlem  28660