Theorem z12shalf 28651

Description: Half of a dyadic is a dyadic. (Contributed by Scott Fenton, 11-Dec-2025.)

Assertion

Ref	Expression
z12shalf	⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])

Proof of Theorem z12shalf

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

Step	Hyp	Ref	Expression
1		elz12s 28643	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)))
2		2no 28590	. . . . . . . 8 ⊢ 2s ∈ No
3		exps1 28599	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
5	4	oveq2i 7423	. . . . . 6 ⊢ ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = ((𝑎 /su (2s↑s𝑛)) /su 2s)
6		zno 28553	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ No )
7	6	adantr 485	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑎 ∈ No )
8		simpr 489	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑛 ∈ ℕ0s)
9		1n0s 28519	. . . . . . . . . 10 ⊢ 1s ∈ ℕ0s
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 1s ∈ ℕ0s)
11	7, 8, 10	pw2divscan4d 28615	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s 𝑎) /su (2s↑s(𝑛 +s 1s ))))
12	4, 2	eqeltri 2859	. . . . . . . . . 10 ⊢ (2s↑s 1s ) ∈ No
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (2s↑s 1s ) ∈ No )
14		peano2n0s 28501	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
15	14	adantl 486	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑛 +s 1s ) ∈ ℕ0s)
16	13, 7, 15	pw2divsassd 28614	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (((2s↑s 1s ) ·s 𝑎) /su (2s↑s(𝑛 +s 1s ))) = ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))))
17	11, 16	eqtr2d 2799	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))) = (𝑎 /su (2s↑s𝑛)))
18	7, 8	pw2divscld 28610	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) ∈ No )
19	7, 15	pw2divscld 28610	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ No )
20	18, 19, 10	pw2divmulsd 28611	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = (𝑎 /su (2s↑s(𝑛 +s 1s ))) ↔ ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))) = (𝑎 /su (2s↑s𝑛))))
21	17, 20	mpbird 260	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
22	5, 21	eqtr3id 2812	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
23		oveq1 7419	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s𝑚)))
24	23	eqeq2d 2774	. . . . . . 7 ⊢ (𝑏 = 𝑎 → ((𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)) ↔ (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑎 /su (2s↑s𝑚))))
25		oveq2 7420	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
26	25	oveq2d 7428	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
27	26	eqeq2d 2774	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑎 /su (2s↑s𝑚)) ↔ (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑎 /su (2s↑s(𝑛 +s 1s )))))
28		simpl 487	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑎 ∈ ℤs)
29		eqidd 2764	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
30	24, 27, 28, 15, 29	2rspcedvdw 3596	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ∃𝑏 ∈ ℤs ∃𝑚 ∈ ℕ0s (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)))
31		elz12s 28643	. . . . . 6 ⊢ ((𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ ℤs[1/2] ↔ ∃𝑏 ∈ ℤs ∃𝑚 ∈ ℕ0s (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)))
32	30, 31	sylibr 237	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ ℤs[1/2])
33	22, 32	eqeltrd 2863	. . . 4 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) ∈ ℤs[1/2])
34		oveq1 7419	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) = ((𝑎 /su (2s↑s𝑛)) /su 2s))
35	34	eleq1d 2848	. . . 4 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑛)) → ((𝐴 /su 2s) ∈ ℤs[1/2] ↔ ((𝑎 /su (2s↑s𝑛)) /su 2s) ∈ ℤs[1/2]))
36	33, 35	syl5ibrcom 250	. . 3 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) ∈ ℤs[1/2]))
37	36	rexlimivv 3207	. 2 ⊢ (∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) ∈ ℤs[1/2])
38	1, 37	sylbi 220	1 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  (class class class)co 7412   No csur 27782   1_s c1s 27977   +_s cadds 28130   ·_s cmuls 28277   /_su cdivs 28358  ℕ_0scn0s 28483  ℤ_sczs 28549  2_sc2s 28581  ↑_scexps 28583  ℤ_s[1/2]cz12s 28585

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-oadd 8458  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-1s 27979  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-norec2 28120  df-adds 28131  df-negs 28192  df-subs 28193  df-muls 28278  df-divs 28359  df-seqs 28455  df-n0s 28485  df-nns 28486  df-zs 28550  df-2s 28582  df-exps 28584  df-z12s 28586

This theorem is referenced by: (None)