Theorem z12negscl 28548

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

Assertion

Ref	Expression
z12negscl	⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])

Proof of Theorem z12negscl

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

Step	Hyp	Ref	Expression
1		oveq1 7399	. . . . . . 7 ⊢ (𝑧 = ( -us ‘𝑥) → (𝑧 /su (2s↑s𝑦)) = (( -us ‘𝑥) /su (2s↑s𝑦)))
2	1	eqeq2d 2772	. . . . . 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 28462	. . . . . . 7 ⊢ (𝑥 ∈ ℤs → ( -us ‘𝑥) ∈ ℤs)
4	3	adantl 485	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘𝑥) ∈ ℤs)
5		zno 28452	. . . . . . . 8 ⊢ (𝑥 ∈ ℤs → 𝑥 ∈ No )
6	5	adantl 485	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → 𝑥 ∈ No )
7		simpl 486	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → 𝑦 ∈ ℕ0s)
8	6, 7	pw2divsnegd 28519	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ( -us ‘(𝑥 /su (2s↑s𝑦))) = (( -us ‘𝑥) /su (2s↑s𝑦)))
9	2, 4, 8	rspcedvdw 3584	. . . . 5 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → ∃𝑧 ∈ ℤs ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦)))
10		fveq2 6863	. . . . . . 7 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → ( -us ‘𝐴) = ( -us ‘(𝑥 /su (2s↑s𝑦))))
11	10	eqeq1d 2763	. . . . . 6 ⊢ (𝐴 = (𝑥 /su (2s↑s𝑦)) → (( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)) ↔ ( -us ‘(𝑥 /su (2s↑s𝑦))) = (𝑧 /su (2s↑s𝑦))))
12	11	rexbidv 3185	. . . . 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 249	. . . 4 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑥 ∈ ℤs) → (𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦))))
14	13	rexlimdva 3162	. . 3 ⊢ (𝑦 ∈ ℕ0s → (∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦))))
15	14	reximia 3096	. 2 ⊢ (∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)) → ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
16		elz12s 28542	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)))
17		rexcom 3290	. . 3 ⊢ (∃𝑥 ∈ ℤs ∃𝑦 ∈ ℕ0s 𝐴 = (𝑥 /su (2s↑s𝑦)) ↔ ∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)))
18	16, 17	bitri 277	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑦 ∈ ℕ0s ∃𝑥 ∈ ℤs 𝐴 = (𝑥 /su (2s↑s𝑦)))
19		elz12s 28542	. . 3 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑦 ∈ ℕ0s ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
20		rexcom 3290	. . 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 277	. 2 ⊢ (( -us ‘𝐴) ∈ ℤs[1/2] ↔ ∃𝑦 ∈ ℕ0s ∃𝑧 ∈ ℤs ( -us ‘𝐴) = (𝑧 /su (2s↑s𝑦)))
22	15, 18, 21	3imtr4i 294	1 ⊢ (𝐴 ∈ ℤs[1/2] → ( -us ‘𝐴) ∈ ℤs[1/2])

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  ‘cfv 6517  (class class class)co 7392   No csur 27681   -u_s cnegs 28089   /_su cdivs 28257  ℕ_0scn0s 28382  ℤ_sczs 28448  2_sc2s 28480  ↑_scexps 28482  ℤ_s[1/2]cz12s 28484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-se 5599  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-nadd 8631  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-0s 27877  df-1s 27878  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-norec 28008  df-norec2 28019  df-adds 28030  df-negs 28091  df-subs 28092  df-muls 28177  df-divs 28258  df-seqs 28354  df-n0s 28384  df-nns 28385  df-zs 28449  df-2s 28481  df-exps 28483  df-z12s 28485

This theorem is referenced by:  z12subscl  28549  z12negsclb  28551  z12bday  28555