Theorem z12shalf 28631

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 28623	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)))
2		2no 28570	. . . . . . . 8 ⊢ 2s ∈ No
3		exps1 28579	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
5	4	oveq2i 7411	. . . . . 6 ⊢ ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = ((𝑎 /su (2s↑s𝑛)) /su 2s)
6		zno 28533	. . . . . . . . . 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 28499	. . . . . . . . . 10 ⊢ 1s ∈ ℕ0s
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 1s ∈ ℕ0s)
11	7, 8, 10	pw2divscan4d 28595	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s 𝑎) /su (2s↑s(𝑛 +s 1s ))))
12	4, 2	eqeltri 2861	. . . . . . . . . 10 ⊢ (2s↑s 1s ) ∈ No
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (2s↑s 1s ) ∈ No )
14		peano2n0s 28481	. . . . . . . . . 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 28594	. . . . . . . 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 2801	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))) = (𝑎 /su (2s↑s𝑛)))
18	7, 8	pw2divscld 28590	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) ∈ No )
19	7, 15	pw2divscld 28590	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ No )
20	18, 19, 10	pw2divmulsd 28591	. . . . . . 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 2814	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
23		oveq1 7407	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s𝑚)))
24	23	eqeq2d 2776	. . . . . . 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 7408	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
26	25	oveq2d 7416	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
27	26	eqeq2d 2776	. . . . . . 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 2766	. . . . . . 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 3598	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ∃𝑏 ∈ ℤs ∃𝑚 ∈ ℕ0s (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)))
31		elz12s 28623	. . . . . 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 2865	. . . 4 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) ∈ ℤs[1/2])
34		oveq1 7407	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) = ((𝑎 /su (2s↑s𝑛)) /su 2s))
35	34	eleq1d 2850	. . . 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 1563   ∈ wcel 2145  ∃wrex 3089  (class class class)co 7400   No csur 27762   1_s c1s 27957   +_s cadds 28110   ·_s cmuls 28257   /_su cdivs 28338  ℕ_0scn0s 28463  ℤ_sczs 28529  2_sc2s 28561  ↑_scexps 28563  ℤ_s[1/2]cz12s 28565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-muls 28258  df-divs 28339  df-seqs 28435  df-n0s 28465  df-nns 28466  df-zs 28530  df-2s 28562  df-exps 28564  df-z12s 28566

This theorem is referenced by: (None)