Theorem zs12negscl 28455

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 7365	. . . . . . 7 ⊢ (𝑧 = ( -us ‘𝑥) → (𝑧 /su (2s↑s𝑦)) = (( -us ‘𝑥) /su (2s↑s𝑦)))
2	1	eqeq2d 2746	. . . . . 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 28369	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ ℤs)
4	3	adantl 481	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘𝑥) ∈ ℤs)
5		zno 28359	. . . . . . . 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 28426	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘(𝑥 /su (2s↑s𝑦))) = (( -us ‘𝑥) /su (2s↑s𝑦)))
9	2, 4, 8	rspcedvdw 3578	. . . . 5 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ∃𝑧 ∈ ℤs ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦)))
10		fveq2 6833	. . . . . . 7 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → ( -us ‘𝐴) = ( -us ‘(𝑥 /su (2s↑s𝑦))))
11	10	eqeq1d 2737	. . . . . 6 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → (( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)) ↔ ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦))))
12	11	rexbidv 3159	. . . . 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 3136	. . 3 ⊢ (𝑦 ∈ ℕ0s → (∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦))))
15	14	reximia 3070	. 2 ⊢ (∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
16		elzs12 28449	. . 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 28449	. . 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 1542   ∈ wcel 2114  ∃wrex 3059  ‘cfv 6491  (class class class)co 7358   No csur 27609   -u_s cnegs 27999   /_su cdivs 28167  ℕ_0scnn0s 28291  ℤ_sczs 28355  2_sc2s 28387  ↑_scexps 28389  ℤ_s[1/2]czs12 28391

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 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-ot 4588  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-nadd 8594  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-0s 27803  df-1s 27804  df-made 27823  df-old 27824  df-left 27826  df-right 27827  df-norec 27918  df-norec2 27929  df-adds 27940  df-negs 28001  df-subs 28002  df-muls 28087  df-divs 28168  df-seqs 28263  df-n0s 28293  df-nns 28294  df-zs 28356  df-2s 28388  df-exps 28390  df-zs12 28392

This theorem is referenced by:  zs12subscl  28456  zs12negsclb  28458  zs12bday  28462