Theorem z12sge0 28654

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 782	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℤs)
2		simpllr 787	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s 𝐴)
3		simprr 784	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
4	2, 3	breqtrd 5138	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s (𝑧 /su (2s↑s𝑝)))
5	1	znod 28554	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ No )
6		simplr 780	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑝 ∈ ℕ0s)
7	5, 6	pw2ge0divsd 28617	. . . . . . . 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 28571	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s ↔ (𝑧 ∈ ℤs ∧ 0s ≤s 𝑧))
10	1, 8, 9	sylanbrc 594	. . . . . 6 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℕ0s)
11		simpr 489	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ ℕ0s)
12		2nns 28589	. . . . . . . . . . . . 13 ⊢ 2s ∈ ℕs
13		simplr 780	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑝 ∈ ℕ0s)
14		nnexpscl 28604	. . . . . . . . . . . . 13 ⊢ ((2s ∈ ℕs ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
15	12, 13, 14	sylancr 598	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
16		eucliddivs 28547	. . . . . . . . . . . 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 595	. . . . . . . . . . 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 28590	. . . . . . . . . . . . . . . . . . 19 ⊢ 2s ∈ No
19		simpllr 787	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
20		expscl 28602	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ No )
21	18, 19, 20	sylancr 598	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No )
22		simprl 782	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
23	22	n0nod 28496	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No )
24		simprr 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
25	24	n0nod 28496	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No )
26	25, 19	pw2divscld 28610	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
27	21, 23, 26	addsdid 28327	. . . . . . . . . . . . . . . . 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 28613	. . . . . . . . . . . . . . . . . 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 7428	. . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ ℕ0s)
35	34	n0nod 28496	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ No )
36	23, 26	addscld 28151	. . . . . . . . . . . . . . 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 28611	. . . . . . . . . . . . . 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 642	. . . . . . . . . . . 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 3228	. . . . . . . . . . 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 728	. . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
44	43	eqeq1d 2765	. . . . . . . . . . 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 642	. . . . . . . . . 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 3230	. . . . . . . . 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 461	. . . . . . 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 728	. . . . . 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 3167	. . . 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 7419	. . . . . . . . 9 ⊢ (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) → (𝑧 /su (2s↑s𝑝)) = ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝)))
53	52	eqeq2d 2774	. . . . . . . 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 28498	. . . . . . . . . . . . 13 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
55	12, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕ0s
56		simplr 780	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
57		n0expscl 28603	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕ0s)
58	55, 56, 57	sylancr 598	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ ℕ0s)
59		simprl 782	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
60		n0mulscl 28516	. . . . . . . . . . 11 ⊢ (((2s↑s𝑝) ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
61	58, 59, 60	syl2anc 595	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
62		simprr 784	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
63		n0addscl 28515	. . . . . . . . . 10 ⊢ ((((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
64	61, 62, 63	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
65	64	n0zsd 28561	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℤs)
66	59	n0nod 28496	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No )
67	66, 56	pw2divscan3d 28612	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)) = 𝑥)
68	67	eqcomd 2769	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 = (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)))
69	68	oveq1d 7427	. . . . . . . . 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 598	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No )
71	70, 66	mulscld 28306	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ No )
72	62	n0nod 28496	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No )
73	71, 72, 56	pw2divsdird 28619	. . . . . . . . 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 2801	. . . . . . . 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 3585	. . . . . . 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 2767	. . . . . . . 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 3189	. . . . . . 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 496	. . . . 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 3222	. . . 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 3187	. 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 28643	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑝 ∈ ℕ0s 𝐴 = (𝑧 /su (2s↑s𝑝)))
84		rexcom 3294	. . 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 3294	. . . 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 3112	. . 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 3294	. . 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))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   class class class wbr 5110  (class class class)co 7412   No csur 27782   <s clts 27783   ≤s cles 27886   0_s c0s 27976   +_s cadds 28130   ·_s cmuls 28277   /_su cdivs 28358  ℕ_0scn0s 28483  ℕ_scnns 28484  ℤ_sczs 28549  2_sc2s 28581  ↑_scexps 28583  ℤ_s[1/2]cz12s 28585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-1s 27979  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-norec2 28120  df-adds 28131  df-negs 28192  df-subs 28193  df-muls 28278  df-divs 28359  df-seqs 28455  df-n0s 28485  df-nns 28486  df-zs 28550  df-2s 28582  df-exps 28584  df-z12s 28586

This theorem is referenced by:  z12bdaylem  28655