Theorem z12shalf 28497

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 28489	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)))
2		2no 28436	. . . . . . . 8 ⊢ 2s ∈ No
3		exps1 28445	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
5	4	oveq2i 7374	. . . . . 6 ⊢ ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = ((𝑎 /su (2s↑s𝑛)) /su 2s)
6		zno 28399	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ No )
7	6	adantr 481	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑎 ∈ No )
8		simpr 485	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑛 ∈ ℕ0s)
9		1n0s 28365	. . . . . . . . . 10 ⊢ 1s ∈ ℕ0s
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 1s ∈ ℕ0s)
11	7, 8, 10	pw2divscan4d 28461	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s 𝑎) /su (2s↑s(𝑛 +s 1s ))))
12	4, 2	eqeltri 2836	. . . . . . . . . 10 ⊢ (2s↑s 1s ) ∈ No
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (2s↑s 1s ) ∈ No )
14		peano2n0s 28347	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
15	14	adantl 482	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑛 +s 1s ) ∈ ℕ0s)
16	13, 7, 15	pw2divsassd 28460	. . . . . . . 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 2776	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))) = (𝑎 /su (2s↑s𝑛)))
18	7, 8	pw2divscld 28456	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) ∈ No )
19	7, 15	pw2divscld 28456	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ No )
20	18, 19, 10	pw2divmulsd 28457	. . . . . . 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 258	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
22	5, 21	eqtr3id 2789	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
23		oveq1 7370	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s𝑚)))
24	23	eqeq2d 2751	. . . . . . 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 7371	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
26	25	oveq2d 7379	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
27	26	eqeq2d 2751	. . . . . . 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 483	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑎 ∈ ℤs)
29		eqidd 2741	. . . . . . 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 3581	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ∃𝑏 ∈ ℤs ∃𝑚 ∈ ℕ0s (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)))
31		elz12s 28489	. . . . . 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 235	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ ℤs[1/2])
33	22, 32	eqeltrd 2840	. . . 4 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) ∈ ℤs[1/2])
34		oveq1 7370	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) = ((𝑎 /su (2s↑s𝑛)) /su 2s))
35	34	eleq1d 2825	. . . 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 248	. . 3 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) ∈ ℤs[1/2]))
37	36	rexlimivv 3182	. 2 ⊢ (∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) ∈ ℤs[1/2])
38	1, 37	sylbi 218	1 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∃wrex 3064  (class class class)co 7363   No csur 27628   1_s c1s 27823   +_s cadds 27976   ·_s cmuls 28123   /_su cdivs 28204  ℕ_0scn0s 28329  ℤ_sczs 28395  2_sc2s 28427  ↑_scexps 28429  ℤ_s[1/2]cz12s 28431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-oadd 8406  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-muls 28124  df-divs 28205  df-seqs 28301  df-n0s 28331  df-nns 28332  df-zs 28396  df-2s 28428  df-exps 28430  df-z12s 28432

This theorem is referenced by: (None)