Theorem z12sge0 28491

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 771	. . . . . . 7 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ ℤs)
2		simpllr 776	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s 𝐴)
3		simprr 773	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
4	2, 3	breqtrd 5126	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 0s ≤s (𝑧 /su (2s↑s𝑝)))
5	1	znod 28391	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑧 ∈ No )
6		simplr 769	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑧 ∈ ℤs ∧ 𝐴 = (𝑧 /su (2s↑s𝑝)))) → 𝑝 ∈ ℕ0s)
7	5, 6	pw2ge0divsd 28454	. . . . . . . 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 28408	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s ↔ (𝑧 ∈ ℤs ∧ 0s ≤s 𝑧))
10	1, 8, 9	sylanbrc 584	. . . . . 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 28426	. . . . . . . . . . . . 13 ⊢ 2s ∈ ℕs
13		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → 𝑝 ∈ ℕ0s)
14		nnexpscl 28441	. . . . . . . . . . . . 13 ⊢ ((2s ∈ ℕs ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
15	12, 13, 14	sylancr 588	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕs)
16		eucliddivs 28384	. . . . . . . . . . . 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 585	. . . . . . . . . . 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 28427	. . . . . . . . . . . . . . . . . . 19 ⊢ 2s ∈ No
19		simpllr 776	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
20		expscl 28439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ No )
21	18, 19, 20	sylancr 588	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No )
22		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
23	22	n0nod 28333	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No )
24		simprr 773	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
25	24	n0nod 28333	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No )
26	25, 19	pw2divscld 28447	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (𝑦 /su (2s↑s𝑝)) ∈ No )
27	21, 23, 26	addsdid 28164	. . . . . . . . . . . . . . . . 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 28450	. . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ ℕ0s)
35	34	n0nod 28333	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ 𝑧 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑧 ∈ No )
36	23, 26	addscld 27988	. . . . . . . . . . . . . . 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 28448	. . . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 3201	. . . . . . . . . . 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 717	. . . . . . . . 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 771	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝐴 = (𝑧 /su (2s↑s𝑝)) ∧ 𝑧 ∈ ℕ0s)) → 𝐴 = (𝑧 /su (2s↑s𝑝)))
44	43	eqeq1d 2739	. . . . . . . . . . 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 632	. . . . . . . . . 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 3203	. . . . . . . . 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 717	. . . . . 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 3140	. . . 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 7375	. . . . . . . . 9 ⊢ (𝑧 = (((2s↑s𝑝) ·s 𝑥) +s 𝑦) → (𝑧 /su (2s↑s𝑝)) = ((((2s↑s𝑝) ·s 𝑥) +s 𝑦) /su (2s↑s𝑝)))
53	52	eqeq2d 2748	. . . . . . . 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 28335	. . . . . . . . . . . . 13 ⊢ (2s ∈ ℕs → 2s ∈ ℕ0s)
55	12, 54	ax-mp 5	. . . . . . . . . . . 12 ⊢ 2s ∈ ℕ0s
56		simplr 769	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑝 ∈ ℕ0s)
57		n0expscl 28440	. . . . . . . . . . . 12 ⊢ ((2s ∈ ℕ0s ∧ 𝑝 ∈ ℕ0s) → (2s↑s𝑝) ∈ ℕ0s)
58	55, 56, 57	sylancr 588	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ ℕ0s)
59		simprl 771	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ ℕ0s)
60		n0mulscl 28353	. . . . . . . . . . 11 ⊢ (((2s↑s𝑝) ∈ ℕ0s ∧ 𝑥 ∈ ℕ0s) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
61	58, 59, 60	syl2anc 585	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s)
62		simprr 773	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ ℕ0s)
63		n0addscl 28352	. . . . . . . . . 10 ⊢ ((((2s↑s𝑝) ·s 𝑥) ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
64	61, 62, 63	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℕ0s)
65	64	n0zsd 28398	. . . . . . . 8 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) +s 𝑦) ∈ ℤs)
66	59	n0nod 28333	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 ∈ No )
67	66, 56	pw2divscan3d 28449	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)) = 𝑥)
68	67	eqcomd 2743	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑥 = (((2s↑s𝑝) ·s 𝑥) /su (2s↑s𝑝)))
69	68	oveq1d 7383	. . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → (2s↑s𝑝) ∈ No )
71	70, 66	mulscld 28143	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → ((2s↑s𝑝) ·s 𝑥) ∈ No )
72	62	n0nod 28333	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ No ∧ 0s ≤s 𝐴) ∧ 𝑝 ∈ ℕ0s) ∧ (𝑥 ∈ ℕ0s ∧ 𝑦 ∈ ℕ0s)) → 𝑦 ∈ No )
73	71, 72, 56	pw2divsdird 28456	. . . . . . . . 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 2775	. . . . . . . 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 3581	. . . . . . 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 2741	. . . . . . . 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 3162	. . . . . . 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 3195	. . . 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 3160	. 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 28480	. . 3 ⊢ (𝐴 ∈ ℤs[1/2] ↔ ∃𝑧 ∈ ℤs ∃𝑝 ∈ ℕ0s 𝐴 = (𝑧 /su (2s↑s𝑝)))
84		rexcom 3267	. . 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 3267	. . . 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 3085	. . 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 3267	. . 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 1542   ∈ wcel 2114  ∃wrex 3062   class class class wbr 5100  (class class class)co 7368   No csur 27619   <s clts 27620   ≤s cles 27724   0_s c0s 27813   +_s cadds 27967   ·_s cmuls 28114   /_su cdivs 28195  ℕ_0scn0s 28320  ℕ_scnns 28321  ℤ_sczs 28386  2_sc2s 28418  ↑_scexps 28420  ℤ_s[1/2]cz12s 28422

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 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-nadd 8604  df-no 27622  df-lts 27623  df-bday 27624  df-les 27725  df-slts 27766  df-cuts 27768  df-0s 27815  df-1s 27816  df-made 27835  df-old 27836  df-left 27838  df-right 27839  df-norec 27946  df-norec2 27957  df-adds 27968  df-negs 28029  df-subs 28030  df-muls 28115  df-divs 28196  df-seqs 28292  df-n0s 28322  df-nns 28323  df-zs 28387  df-2s 28419  df-exps 28421  df-z12s 28423

This theorem is referenced by:  z12bdaylem  28492