Theorem zs12negscl 28359

Description: The dyadics are closed under negation. (Contributed by Scott Fenton, 9-Nov-2025.)

Assertion

Ref	Expression
zs12negscl	⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])

Proof of Theorem zs12negscl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7356	. . . . . . 7 ⊢ (𝑧 = ( -us ‘𝑥) → (𝑧 /su (2s↑s𝑦)) = (( -us ‘𝑥) /su (2s↑s𝑦)))
2	1	eqeq2d 2740	. . . . . 6 ⊢ (𝑧 = ( -us ‘𝑥) → (( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦)) ↔ ( -us ‘(𝑥 /su (2s↑s𝑦))) = (( -us ‘𝑥) /su (2s↑s𝑦))))
3		znegscl 28287	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ ℤs)
4	3	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘𝑥) ∈ ℤs)
5		zno 28277	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ No )
6	5	adantl 481	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → 𝑥 ∈ No )
7		simpl 482	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → 𝑦 ∈ ℕ0s)
8	6, 7	pw2divsnegd 28343	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘(𝑥 /su (2s↑s𝑦))) = (( -us ‘𝑥) /su (2s↑s𝑦)))
9	2, 4, 8	rspcedvdw 3580	. . . . 5 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ∃𝑧 ∈ ℤs ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦)))
10		fveq2 6822	. . . . . . 7 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → ( -us ‘𝐴) = ( -us ‘(𝑥 /su (2s↑s𝑦))))
11	10	eqeq1d 2731	. . . . . 6 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → (( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)) ↔ ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦))))
12	11	rexbidv 3153	. . . . 5 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → (∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)) ↔ ∃𝑧 ∈ ℤs ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦))))
13	9, 12	syl5ibrcom 247	. . . 4 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → (𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦))))
14	13	rexlimdva 3130	. . 3 ⊢ (𝑦 ∈ ℕ0s → (∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦))))
15	14	reximia 3064	. 2 ⊢ (∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
16		elzs12 28354	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
17		rexcom 3258	. . 3 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)))
18	16, 17	bitri 275	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)))
19		elzs12 28354	. . 3 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑦 ∈ ℕ0s ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
20		rexcom 3258	. . 3 ⊢ (∃𝑧 ∈ ℤs ∃𝑦 ∈ ℕ0s ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)) ↔ ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
21	19, 20	bitri 275	. 2 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] ↔ ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
22	15, 18, 21	3imtr4i 292	1 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ‘cfv 6482  (class class class)co 7349   No csur 27549   -u_s cnegs 27932   /_su cdivs 28097  ℕ_0scnn0s 28213  ℤ_sczs 28273  2_sc2s 28304  ↑_scexps 28306  ℤ_s[1/2]czs12 28308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-left 27762  df-right 27763  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017  df-divs 28098  df-seqs 28185  df-n0s 28215  df-nns 28216  df-zs 28274  df-2s 28305  df-exps 28307  df-zs12 28309

This theorem is referenced by:  zs12subscl  28360  zs12negsclb  28362