Theorem zs12negscl 28391

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 7376	. . . . . . 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 28321	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ ℤs)
4	3	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘𝑥) ∈ ℤs)
5		zno 28311	. . . . . . . 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 28377	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘(𝑥 /su (2s↑s𝑦))) = (( -us ‘𝑥) /su (2s↑s𝑦)))
9	2, 4, 8	rspcedvdw 3588	. . . . 5 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ∃𝑧 ∈ ℤs ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦)))
10		fveq2 6840	. . . . . . 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 3157	. . . . 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 3134	. . 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 28386	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
17		rexcom 3264	. . 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 28386	. . 3 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑦 ∈ ℕ0s ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
20		rexcom 3264	. . 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 6499  (class class class)co 7369   No csur 27585   -u_s cnegs 27966   /_su cdivs 28131  ℕ_0scnn0s 28247  ℤ_sczs 28307  2_sc2s 28338  ↑_scexps 28340  ℤ_s[1/2]czs12 28342

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 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-oadd 8415  df-nadd 8607  df-no 27588  df-slt 27589  df-bday 27590  df-sle 27691  df-sslt 27728  df-scut 27730  df-0s 27774  df-1s 27775  df-made 27793  df-old 27794  df-left 27796  df-right 27797  df-norec 27886  df-norec2 27897  df-adds 27908  df-negs 27968  df-subs 27969  df-muls 28051  df-divs 28132  df-seqs 28219  df-n0s 28249  df-nns 28250  df-zs 28308  df-2s 28339  df-exps 28341  df-zs12 28343

This theorem is referenced by:  zs12subscl  28392  zs12negsclb  28394