Theorem zs12half 28390

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

Assertion

Ref	Expression
zs12half	⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])

Proof of Theorem zs12half

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

Step	Hyp	Ref	Expression
1		elzs12 28383	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)))
2		2sno 28342	. . . . . . . 8 ⊢ 2s ∈ No
3		exps1 28351	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
5	4	oveq2i 7357	. . . . . 6 ⊢ ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = ((𝑎 /su (2s↑s𝑛)) /su 2s)
6		zno 28306	. . . . . . . . . 10 ⊢ (𝑎 ∈ ℤs → 𝑎 ∈ No )
7	6	adantr 480	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑎 ∈ No )
8		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑛 ∈ ℕ0s)
9		1n0s 28276	. . . . . . . . . 10 ⊢ 1s ∈ ℕ0s
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 1s ∈ ℕ0s)
11	7, 8, 10	pw2divscan4d 28367	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s 𝑎) /su (2s↑s(𝑛 +s 1s ))))
12	4, 2	eqeltri 2827	. . . . . . . . . 10 ⊢ (2s↑s 1s ) ∈ No
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (2s↑s 1s ) ∈ No )
14		peano2n0s 28259	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
15	14	adantl 481	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑛 +s 1s ) ∈ ℕ0s)
16	13, 7, 15	pw2divsassd 28366	. . . . . . . 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 2767	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))) = (𝑎 /su (2s↑s𝑛)))
18	7, 8	pw2divscld 28362	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) ∈ No )
19	7, 15	pw2divscld 28362	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ No )
20	18, 19, 10	pw2divsmuld 28363	. . . . . . 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 257	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
22	5, 21	eqtr3id 2780	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
23		oveq1 7353	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s𝑚)))
24	23	eqeq2d 2742	. . . . . . 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 7354	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
26	25	oveq2d 7362	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
27	26	eqeq2d 2742	. . . . . . 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 482	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 𝑎 ∈ ℤs)
29		eqidd 2732	. . . . . . 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 3586	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ∃𝑏 ∈ ℤs ∃𝑚 ∈ ℕ0s (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)))
31		elzs12 28383	. . . . . 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 234	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ ℤs[1/2])
33	22, 32	eqeltrd 2831	. . . 4 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) ∈ ℤs[1/2])
34		oveq1 7353	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) = ((𝑎 /su (2s↑s𝑛)) /su 2s))
35	34	eleq1d 2816	. . . 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 247	. . 3 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) ∈ ℤs[1/2]))
37	36	rexlimivv 3174	. 2 ⊢ (∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) ∈ ℤs[1/2])
38	1, 37	sylbi 217	1 ⊢ (𝐴 ∈ ℤs[1/2] → (𝐴 /su 2s) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  (class class class)co 7346   No csur 27578   1_s c1s 27767   +_s cadds 27902   ·_s cmuls 28045   /_su cdivs 28126  ℕ_0scnn0s 28242  ℤ_sczs 28302  2_sc2s 28333  ↑_scexps 28335  ℤ_s[1/2]czs12 28337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-oadd 8389  df-nadd 8581  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-0s 27768  df-1s 27769  df-made 27788  df-old 27789  df-left 27791  df-right 27792  df-norec 27881  df-norec2 27892  df-adds 27903  df-negs 27963  df-subs 27964  df-muls 28046  df-divs 28127  df-seqs 28214  df-n0s 28244  df-nns 28245  df-zs 28303  df-2s 28334  df-exps 28336  df-zs12 28338

This theorem is referenced by: (None)