Theorem z12shalf 28486

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 28478	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑎 ∈ ℤs ∃𝑛 ∈ ℕ0s 𝐴 = (𝑎 /su (2s↑s𝑛)))
2		2no 28425	. . . . . . . 8 ⊢ 2s ∈ No
3		exps1 28434	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
5	4	oveq2i 7371	. . . . . 6 ⊢ ((𝑎 /su (2s↑s𝑛)) /su (2s↑s 1s )) = ((𝑎 /su (2s↑s𝑛)) /su 2s)
6		zno 28388	. . . . . . . . . 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 28354	. . . . . . . . . 10 ⊢ 1s ∈ ℕ0s
10	9	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → 1s ∈ ℕ0s)
11	7, 8, 10	pw2divscan4d 28450	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s 𝑎) /su (2s↑s(𝑛 +s 1s ))))
12	4, 2	eqeltri 2833	. . . . . . . . . 10 ⊢ (2s↑s 1s ) ∈ No
13	12	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (2s↑s 1s ) ∈ No )
14		peano2n0s 28336	. . . . . . . . . 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 28449	. . . . . . . 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 2773	. . . . . . 7 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((2s↑s 1s ) ·s (𝑎 /su (2s↑s(𝑛 +s 1s )))) = (𝑎 /su (2s↑s𝑛)))
18	7, 8	pw2divscld 28445	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s𝑛)) ∈ No )
19	7, 15	pw2divscld 28445	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → (𝑎 /su (2s↑s(𝑛 +s 1s ))) ∈ No )
20	18, 19, 10	pw2divmulsd 28446	. . . . . . 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 2786	. . . . 5 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
23		oveq1 7367	. . . . . . . 8 ⊢ (𝑏 = 𝑎 → (𝑏 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s𝑚)))
24	23	eqeq2d 2748	. . . . . . 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 7368	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
26	25	oveq2d 7376	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑎 /su (2s↑s𝑚)) = (𝑎 /su (2s↑s(𝑛 +s 1s ))))
27	26	eqeq2d 2748	. . . . . . 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 2738	. . . . . . 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 3579	. . . . . 6 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ∃𝑏 ∈ ℤs ∃𝑚 ∈ ℕ0s (𝑎 /su (2s↑s(𝑛 +s 1s ))) = (𝑏 /su (2s↑s𝑚)))
31		elz12s 28478	. . . . . 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 2837	. . . 4 ⊢ ((𝑎 ∈ ℤs ∧ 𝑛 ∈ ℕ0s) → ((𝑎 /su (2s↑s𝑛)) /su 2s) ∈ ℤs[1/2])
34		oveq1 7367	. . . . 5 ⊢ (𝐴 = (𝑎 /su (2s↑s𝑛)) → (𝐴 /su 2s) = ((𝑎 /su (2s↑s𝑛)) /su 2s))
35	34	eleq1d 2822	. . . 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 3180	. 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 1542   ∈ wcel 2114  ∃wrex 3062  (class class class)co 7360   No csur 27617   1_s c1s 27812   +_s cadds 27965   ·_s cmuls 28112   /_su cdivs 28193  ℕ_0scn0s 28318  ℤ_sczs 28384  2_sc2s 28416  ↑_scexps 28418  ℤ_s[1/2]cz12s 28420

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-ot 4577  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-les 27723  df-slts 27764  df-cuts 27766  df-0s 27813  df-1s 27814  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028  df-muls 28113  df-divs 28194  df-seqs 28290  df-n0s 28320  df-nns 28321  df-zs 28385  df-2s 28417  df-exps 28419  df-z12s 28421

This theorem is referenced by: (None)