Theorem zs12ge0 28396

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

Assertion

Ref	Expression
zs12ge0	⊢ ((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 zs12ge0

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 770	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℤs)
2		simpllr 775	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s 𝐴)
3		simprr 772	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
4	2, 3	breqtrd 5128	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s (𝑧 /su (2s↑s𝑝)))
5	1	znod 28312	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ No )
6		simplr 768	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑝 ∈ ℕ0s)
7	5, 6	pw2ge0divsd 28374	. . . . . . . 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 257	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s 𝑧)
9		eln0zs 28329	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s ↔ (𝑧 ∈ ℤs ∧ 0s ≤s 𝑧))
10	1, 8, 9	sylanbrc 583	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℕ0s)
11		simpr 484	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ ℕ0s)
12		2nns 28346	. . . . . . . . . . . . 13 ⊢ 2s ∈ ℕs
13		simplr 768	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑝 ∈ ℕ0s)
14		nnexpscl 28361	. . . . . . . . . . . . 13 ⊢ ((2s ∈ ℕs ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
15	12, 13, 14	sylancr 587	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
16		eucliddivs 28306	. . . . . . . . . . . 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 584	. . . . . . . . . . 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		2sno 28347	. . . . . . . . . . . . . . . . . . 19 ⊢ 2s ∈ No
19		simpllr 775	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
20		expscl 28359	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ No )
21	18, 19, 20	sylancr 587	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No )
22		simprl 770	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
23	22	n0snod 28259	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No )
24		simprr 772	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
25	24	n0snod 28259	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No )
26	25, 19	pw2divscld 28367	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
27	21, 23, 26	addsdid 28100	. . . . . . . . . . . . . . . . 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 28370	. . . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ ℕ0s)
35	34	n0snod 28259	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ No )
36	23, 26	addscld 27928	. . . . . . . . . . . . . . 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	pw2divsmuld 28368	. . . . . . . . . . . . . 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 282	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 3198	. . . . . . . . . . 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 232	. . . . . . . . . 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 716	. . . . . . . . 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 770	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
44	43	eqeq1d 2731	. . . . . . . . . . 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 631	. . . . . . . . . 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 3200	. . . . . . . . 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 257	. . . . . . . 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 456	. . . . . . 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 716	. . . . . 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 15	. . . . 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 3135	. . . 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 7376	. . . . . . . . 9 ⊢ (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) → (𝑧 /su (2s↑s𝑝)) = ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝)))
53	52	eqeq2d 2740	. . . . . . . 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 28261	. . . . . . . . . . . . 13 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
55	12, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕ0s
56		simplr 768	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
57		n0expscl 28360	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕ0s)
58	55, 56, 57	sylancr 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ ℕ0s)
59		simprl 770	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
60		n0mulscl 28278	. . . . . . . . . . 11 ⊢ (((2s↑s𝑝) ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
61	58, 59, 60	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
62		simprr 772	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
63		n0addscl 28277	. . . . . . . . . 10 ⊢ ((((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
64	61, 62, 63	syl2anc 584	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
65	64	n0zsd 28319	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℤs)
66	59	n0snod 28259	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No )
67	66, 56	pw2divscan3d 28369	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)) = 𝑥)
68	67	eqcomd 2735	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 = (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)))
69	68	oveq1d 7384	. . . . . . . . 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 587	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No )
71	70, 66	mulscld 28079	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ No )
72	62	n0snod 28259	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No )
73	71, 72, 56	pw2divsdird 28376	. . . . . . . . 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 2767	. . . . . . . 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 3588	. . . . . . 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 2733	. . . . . . . 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 3157	. . . . . . 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 247	. . . . . 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 491	. . . . 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 3192	. . . 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 212	. . 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 3155	. 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		elzs12 28386	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑝 ∈ ℕ0s 𝐴 = (𝑧 /su (2s↑s𝑝)))
84		rexcom 3264	. . 3 ⊢ (∃𝑧 ∈ ℤs ∃𝑝 ∈ ℕ0s 𝐴 = (𝑧 /su (2s↑s𝑝)) ↔ ∃𝑝 ∈ ℕ0s ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)))
85	83, 84	bitri 275	. 2 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑝 ∈ ℕ0s ∃𝑧 ∈ ℤs 𝐴 = (𝑧 /su (2s↑s𝑝)))
86		rexcom 3264	. . . 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 3076	. . 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 3264	. . 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 275	. 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 314	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))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   class class class wbr 5102  (class class class)co 7369   No csur 27585   <s cslt 27586   ≤s csle 27690   0_s c0s 27772   +_s cadds 27907   ·_s cmuls 28050   /_su cdivs 28131  ℕ_0scnn0s 28247  ℕ_scnns 28248  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: (None)