Theorem zs12negscl 28388

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 7353	. . . . . . 7 ⊢ (𝑧 = ( -us ‘𝑥) → (𝑧 /su (2s↑s𝑦)) = (( -us ‘𝑥) /su (2s↑s𝑦)))
2	1	eqeq2d 2742	. . . . . 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 28316	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ ℤs)
4	3	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘𝑥) ∈ ℤs)
5		zno 28306	. . . . . . . 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 28372	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘(𝑥 /su (2s↑s𝑦))) = (( -us ‘𝑥) /su (2s↑s𝑦)))
9	2, 4, 8	rspcedvdw 3575	. . . . 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 2733	. . . . . 6 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → (( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)) ↔ ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦))))
12	11	rexbidv 3156	. . . . 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 3133	. . 3 ⊢ (𝑦 ∈ ℕ0s → (∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦))))
15	14	reximia 3067	. 2 ⊢ (∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
16		elzs12 28383	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
17		rexcom 3261	. . 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 28383	. . 3 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑦 ∈ ℕ0s ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
20		rexcom 3261	. . 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 1541   ∈ wcel 2111  ∃wrex 3056  ‘cfv 6481  (class class class)co 7346   No csur 27578   -u_s cnegs 27961   /_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:  zs12subscl  28389  zs12negsclb  28391