Theorem z12sge0 28874

Description: An expression for non-negative dyadic rationals. (Contributed by Scott Fenton, 8-Nov-2025.)

Assertion

Ref	Expression
z12sge0	⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))

Distinct variable group:   𝑥,𝐴,𝑦,𝑝

Proof of Theorem z12sge0

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 783	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℤs)
2		simpllr 788	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s 𝐴)
3		simprr 785	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
4	2, 3	breqtrd 5130	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s (𝑧 /su (2s↑s𝑝)))
5	1	znod 28774	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ No)
6		simplr 781	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑝 ∈ ℕ0s)
7	5, 6	pw2ge0divsd 28837	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → ( 0s ≤s 𝑧 ↔ 0s ≤s (𝑧 /su (2s↑s𝑝))))
8	4, 7	mpbird 260	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s 𝑧)
9		eln0zs 28791	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s ↔ (𝑧 ∈ ℤs ∧ 0s ≤s 𝑧))
10	1, 8, 9	sylanbrc 595	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℕ0s)
11		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ ℕ0s)
12		2nns 28809	. . . . . . . . . . . . 13 ⊢ 2s ∈ ℕs
13		simplr 781	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑝 ∈ ℕ0s)
14		nnexpscl 28824	. . . . . . . . . . . . 13 ⊢ ((2s ∈ ℕs ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
15	12, 13, 14	sylancr 599	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
16		eucliddivs 28767	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ℕ0s ∧ (2s↑s𝑝) ∈ ℕs) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∧ 𝑦 <s (2s↑s𝑝)))
17	11, 15, 16	syl2anc 596	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∧ 𝑦 <s (2s↑s𝑝)))
18		2no 28810	. . . . . . . . . . . . . . . . . . 19 ⊢ 2s ∈ No
19		simpllr 788	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
20		expscl 28822	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ No)
21	18, 19, 20	sylancr 599	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No)
22		simprl 783	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
23	22	n0nod 28716	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No)
24		simprr 785	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
25	24	n0nod 28716	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No)
26	25, 19	pw2divscld 28830	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑦 /su (2s↑s𝑝)) ∈ No)
27	21, 23, 26	addsdid 28547	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) = (((2s↑s𝑝) ·s 𝑥) +s ((2s↑s𝑝) ·s (𝑦 /su (2s↑s𝑝)))))
28	25, 19	pw2divscan2d 28833	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s (𝑦 /su (2s↑s𝑝))) = 𝑦)
29	28	oveq2d 7436	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s ((2s↑s𝑝) ·s (𝑦 /su (2s↑s𝑝)))) = (((2s↑s𝑝) ·s 𝑥) +s 𝑦))
30	27, 29	eqtrd 2795	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) = (((2s↑s𝑝) ·s 𝑥) +s 𝑦))
31	30	eqeq2d 2771	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑧 = ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) ↔ 𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦)))
32		eqcom 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) ↔ ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) = 𝑧)
33	31, 32	bitr3di 289	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ↔ ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) = 𝑧))
34		simplr 781	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ ℕ0s)
35	34	n0nod 28716	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ No)
36	23, 26	addscld 28371	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∈ No)
37	35, 36, 19	pw2divmulsd 28831	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ ((2s↑s𝑝) ·s (𝑥 +s (𝑦 /su (2s↑s𝑝)))) = 𝑧))
38	33, 37	bitr4d 285	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ↔ (𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
39	38	anbi1d 643	. . . . . . . . . . . 12 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
40	39	2rexbidva 3225	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
41	17, 40	mpbid 235	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
42	41	adantrl 729	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
43		simprl 783	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
44	43	eqeq1d 2762	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ (𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
45	44	anbi1d 643	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
46	45	2rexbidv 3227	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ((𝑧 /su (2s↑s𝑝)) = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
47	42, 46	mpbird 260	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
48	47	expr 462	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝐴 = (𝑧 /su (2s↑s𝑝))) → (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
49	48	adantrl 729	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → (𝑧 ∈ ℕ0s → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
50	10, 49	mpd 16	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
51	50	rexlimdvaa 3164	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) → (∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)) → ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
52		oveq1 7427	. . . . . . . . 9 ⊢ (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) → (𝑧 /su (2s↑s𝑝)) = ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝)))
53	52	eqeq2d 2771	. . . . . . . 8 ⊢ (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) → ((𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑧 /su (2s↑s𝑝)) ↔ (𝑥 +s (𝑦 /su (2s↑s𝑝))) = ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝))))
54		nnn0s 28718	. . . . . . . . . . . . 13 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
55	12, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕ0s
56		simplr 781	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
57		n0expscl 28823	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕ0s)
58	55, 56, 57	sylancr 599	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ ℕ0s)
59		simprl 783	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
60		n0mulscl 28736	. . . . . . . . . . 11 ⊢ (((2s↑s𝑝) ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
61	58, 59, 60	syl2anc 596	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
62		simprr 785	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
63		n0addscl 28735	. . . . . . . . . 10 ⊢ ((((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
64	61, 62, 63	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
65	64	n0zsd 28781	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℤs)
66	59	n0nod 28716	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No)
67	66, 56	pw2divscan3d 28832	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)) = 𝑥)
68	67	eqcomd 2766	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 = (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)))
69	68	oveq1d 7435	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = ((((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)) +s (𝑦 /su (2s↑s𝑝))))
70	18, 56, 20	sylancr 599	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No)
71	70, 66	mulscld 28526	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ No)
72	62	n0nod 28716	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No)
73	71, 72, 56	pw2divsdird 28839	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝)) = ((((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)) +s (𝑦 /su (2s↑s𝑝))))
74	69, 73	eqtr4d 2798	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝)))
75	53, 65, 74	rspcedvdw 3579	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ∃𝑧 ∈ ℤs (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑧 /su (2s↑s𝑝)))
76		eqeq1 2764	. . . . . . . 8 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → (𝐴 = (𝑧 /su (2s↑s𝑝)) ↔ (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑧 /su (2s↑s𝑝))))
77	76	rexbidv 3186	. . . . . . 7 ⊢ (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → (∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)) ↔ ∃𝑧 ∈ ℤs (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑧 /su (2s↑s𝑝))))
78	75, 77	syl5ibrcom 250	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) → ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝))))
79	78	adantrd 497	. . . . 5 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝))))
80	79	rexlimdvva 3219	. . . 4 ⊢ (((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) → ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝))))
81	51, 80	impbid 215	. . 3 ⊢ (((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) → (∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
82	81	rexbidva 3184	. 2 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (∃𝑝 ∈ ℕ0s ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)) ↔ ∃𝑝 ∈ ℕ0s ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))
83		elz12s 28863	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑝 ∈ ℕ0s 𝐴 = (𝑧 /su (2s↑s𝑝)))
84		rexcom 3291	. . 3 ⊢ (∃𝑧 ∈ ℤs ∃𝑝 ∈ ℕ0s 𝐴 = (𝑧 /su (2s↑s𝑝)) ↔ ∃𝑝 ∈ ℕ0s ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)))
85	83, 84	bitri 278	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑝 ∈ ℕ0s ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)))
86		rexcom 3291	. . . 4 ⊢ (∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ∃𝑝 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
87	86	rexbii 3109	. . 3 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ∃𝑥 ∈ ℕ0s ∃𝑝 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
88		rexcom 3291	. . 3 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑝 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ∃𝑝 ∈ ℕ0s ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
89	87, 88	bitri 278	. 2 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)) ↔ ∃𝑝 ∈ ℕ0s ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝)))
90	82, 85, 89	3bitr4g 317	1 ⊢ ((𝐴 ∈ No ∧ 0s ≤s 𝐴) → (𝐴 ∈ ℤs[1/2] ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝐴 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃wrex 3086   class class class wbr 5102  (class class class)co 7420  Nocsur 28002   <s clts 28003   ≤s cles 28106   0_s c0s 28196   +_s cadds 28350   ·_s cmuls 28497   /_su cdivs 28578  ℕ_0scn0s 28703  ℕ_scnns 28704  ℤ_sczs 28769  2_sc2s 28801  ↑_scexps 28803  ℤ_s[1/2]cz12s 28805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-oadd 8483  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-muls 28498  df-divs 28579  df-seqs 28675  df-n0s 28705  df-nns 28706  df-zs 28770  df-2s 28802  df-exps 28804  df-z12s 28806

This theorem is used by:  z12bdaylem  28875