Theorem pw2cut2 28353

Description: Cut expression for powers of two. Theorem 12 of [Conway] p. 12-13. (Contributed by Scott Fenton, 18-Jan-2026.)

Assertion

Ref	Expression
pw2cut2	⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}))

Proof of Theorem pw2cut2

Dummy variables 𝑎 𝑏 𝑚 𝑛 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7357	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2sno 28313	. . . . . . . 8 ⊢ 2s ∈ No
3		exps0 28321	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2780	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq2d 7365	. . . . 5 ⊢ (𝑚 = 0s → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su 1s ))
7	5	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su 1s ))
8	7	sneqd 4589	. . . . . 6 ⊢ (𝑚 = 0s → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su 1s )})
9	5	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su 1s ))
10	9	sneqd 4589	. . . . . 6 ⊢ (𝑚 = 0s → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su 1s )})
11	8, 10	oveq12d 7367	. . . . 5 ⊢ (𝑚 = 0s → ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) = ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}))
12	6, 11	eqeq12d 2745	. . . 4 ⊢ (𝑚 = 0s → ((𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) ↔ (𝐴 /su 1s ) = ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )})))
13	12	imbi2d 340	. . 3 ⊢ (𝑚 = 0s → ((𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))})) ↔ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}))))
14		oveq2 7357	. . . . . 6 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
15	14	oveq2d 7365	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s𝑛)))
16	14	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s𝑛)))
17	16	sneqd 4589	. . . . . 6 ⊢ (𝑚 = 𝑛 → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s𝑛))})
18	14	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s𝑛)))
19	18	sneqd 4589	. . . . . 6 ⊢ (𝑚 = 𝑛 → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s𝑛))})
20	17, 19	oveq12d 7367	. . . . 5 ⊢ (𝑚 = 𝑛 → ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))}))
21	15, 20	eqeq12d 2745	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) ↔ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})))
22	21	imbi2d 340	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))})) ↔ (𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))}))))
23		oveq2 7357	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
24	23	oveq2d 7365	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s(𝑛 +s 1s ))))
25	23	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))))
26	25	sneqd 4589	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))})
27	23	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))
28	27	sneqd 4589	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
29	26, 28	oveq12d 7367	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
30	24, 29	eqeq12d 2745	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) ↔ (𝐴 /su (2s↑s(𝑛 +s 1s ))) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
31	30	imbi2d 340	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))})) ↔ (𝐴 ∈ ℤs → (𝐴 /su (2s↑s(𝑛 +s 1s ))) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
32		oveq2 7357	. . . . . 6 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
33	32	oveq2d 7365	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s𝑁)))
34	32	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s𝑁)))
35	34	sneqd 4589	. . . . . 6 ⊢ (𝑚 = 𝑁 → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s𝑁))})
36	32	oveq2d 7365	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s𝑁)))
37	36	sneqd 4589	. . . . . 6 ⊢ (𝑚 = 𝑁 → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s𝑁))})
38	35, 37	oveq12d 7367	. . . . 5 ⊢ (𝑚 = 𝑁 → ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}))
39	33, 38	eqeq12d 2745	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))}) ↔ (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))})))
40	39	imbi2d 340	. . 3 ⊢ (𝑚 = 𝑁 → ((𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑚)) = ({((𝐴 -s 1s ) /su (2s↑s𝑚))} |s {((𝐴 +s 1s ) /su (2s↑s𝑚))})) ↔ (𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}))))
41		zscut 28302	. . . 4 ⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
42		zno 28277	. . . . 5 ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
43		divs1 28114	. . . . 5 ⊢ (𝐴 ∈ No → (𝐴 /su 1s ) = 𝐴)
44	42, 43	syl 17	. . . 4 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = 𝐴)
45		1sno 27742	. . . . . . . . 9 ⊢ 1s ∈ No
46	45	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℤs → 1s ∈ No )
47	42, 46	subscld 27974	. . . . . . 7 ⊢ (𝐴 ∈ ℤs → (𝐴 -s 1s ) ∈ No )
48		divs1 28114	. . . . . . 7 ⊢ ((𝐴 -s 1s ) ∈ No → ((𝐴 -s 1s ) /su 1s ) = (𝐴 -s 1s ))
49	47, 48	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤs → ((𝐴 -s 1s ) /su 1s ) = (𝐴 -s 1s ))
50	49	sneqd 4589	. . . . 5 ⊢ (𝐴 ∈ ℤs → {((𝐴 -s 1s ) /su 1s )} = {(𝐴 -s 1s )})
51	42, 46	addscld 27894	. . . . . . 7 ⊢ (𝐴 ∈ ℤs → (𝐴 +s 1s ) ∈ No )
52		divs1 28114	. . . . . . 7 ⊢ ((𝐴 +s 1s ) ∈ No → ((𝐴 +s 1s ) /su 1s ) = (𝐴 +s 1s ))
53	51, 52	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ℤs → ((𝐴 +s 1s ) /su 1s ) = (𝐴 +s 1s ))
54	53	sneqd 4589	. . . . 5 ⊢ (𝐴 ∈ ℤs → {((𝐴 +s 1s ) /su 1s )} = {(𝐴 +s 1s )})
55	50, 54	oveq12d 7367	. . . 4 ⊢ (𝐴 ∈ ℤs → ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
56	41, 44, 55	3eqtr4d 2774	. . 3 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}))
57		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝐴 ∈ ℤs)
58	57	znod 28278	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝐴 ∈ No )
59	45	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 1s ∈ No )
60	58, 59	subscld 27974	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 -s 1s ) ∈ No )
61		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝑛 ∈ ℕ0s)
62		peano2n0s 28230	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
63	61, 62	syl 17	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝑛 +s 1s ) ∈ ℕ0s)
64	60, 63	pw2divscld 28333	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ No )
65	58, 59	addscld 27894	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 +s 1s ) ∈ No )
66	65, 63	pw2divscld 28333	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ No )
67	58	sltm1d 28012	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 -s 1s ) <s 𝐴)
68	58	sltp1d 27929	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝐴 <s (𝐴 +s 1s ))
69	60, 58, 65, 67, 68	slttrd 27669	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 -s 1s ) <s (𝐴 +s 1s ))
70	60, 65, 63	pw2sltdiv1d 28346	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) <s (𝐴 +s 1s ) ↔ ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) <s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
71	69, 70	mpbid 232	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) <s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))
72	64, 66, 71	ssltsn 27704	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} <<s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
73	72	scutcld 27715	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ No )
74	64, 73	addscld 27894	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ∈ No )
75	66, 73	addscld 27894	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ∈ No )
76	64, 66, 73	sltadd1d 27912	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) <s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ↔ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
77	71, 76	mpbid 232	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
78	74, 75, 77	ssltsn 27704	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
79	60, 61	pw2divscld 28333	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s𝑛)) ∈ No )
80	65, 61	pw2divscld 28333	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 +s 1s ) /su (2s↑s𝑛)) ∈ No )
81	60, 65, 61	pw2sltdiv1d 28346	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) <s (𝐴 +s 1s ) ↔ ((𝐴 -s 1s ) /su (2s↑s𝑛)) <s ((𝐴 +s 1s ) /su (2s↑s𝑛))))
82	69, 81	mpbid 232	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s𝑛)) <s ((𝐴 +s 1s ) /su (2s↑s𝑛)))
83	79, 80, 82	ssltsn 27704	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {((𝐴 -s 1s ) /su (2s↑s𝑛))} <<s {((𝐴 +s 1s ) /su (2s↑s𝑛))})
84		eqidd 2730	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} |s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) = ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} |s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
85		simp3 1138	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))}))
86	58, 61	pw2divscld 28333	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 /su (2s↑s𝑛)) ∈ No )
87		scutcut 27713	. . . . . . . . . . . . . 14 ⊢ ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} <<s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))} → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ No ∧ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} <<s {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})} ∧ {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})} <<s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
88	72, 87	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ No ∧ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} <<s {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})} ∧ {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})} <<s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
89	88	simp3d 1144	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})} <<s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
90		ovex 7382	. . . . . . . . . . . . . 14 ⊢ ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ V
91	90	snid 4614	. . . . . . . . . . . . 13 ⊢ ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})}
92	91	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})})
93		ovex 7382	. . . . . . . . . . . . . 14 ⊢ ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ V
94	93	snid 4614	. . . . . . . . . . . . 13 ⊢ ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}
95	94	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
96	89, 92, 95	ssltsepcd 27706	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) <s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))
97	73, 66, 64	sltadd2d 27911	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) <s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ↔ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))))
98	96, 97	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
99	58, 58, 59	addsassd 27920	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 +s 𝐴) +s 1s ) = (𝐴 +s (𝐴 +s 1s )))
100	99	oveq1d 7364	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 𝐴) +s 1s ) -s 1s ) = ((𝐴 +s (𝐴 +s 1s )) -s 1s ))
101	58, 58	addscld 27894	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 +s 𝐴) ∈ No )
102		pncans 27983	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 +s 𝐴) ∈ No ∧ 1s ∈ No ) → (((𝐴 +s 𝐴) +s 1s ) -s 1s ) = (𝐴 +s 𝐴))
103	101, 45, 102	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 𝐴) +s 1s ) -s 1s ) = (𝐴 +s 𝐴))
104		no2times 28311	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
105	58, 104	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s 𝐴) = (𝐴 +s 𝐴))
106	103, 105	eqtr4d 2767	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 𝐴) +s 1s ) -s 1s ) = (2s ·s 𝐴))
107	58, 65, 59	addsubsd 27993	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 +s (𝐴 +s 1s )) -s 1s ) = ((𝐴 -s 1s ) +s (𝐴 +s 1s )))
108	100, 106, 107	3eqtr3rd 2773	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) +s (𝐴 +s 1s )) = (2s ·s 𝐴))
109	108	oveq1d 7364	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) +s (𝐴 +s 1s )) /su (2s↑s(𝑛 +s 1s ))) = ((2s ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))))
110	60, 65, 63	pw2divsdird 28342	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) +s (𝐴 +s 1s )) /su (2s↑s(𝑛 +s 1s ))) = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
111		1n0s 28247	. . . . . . . . . . . . . 14 ⊢ 1s ∈ ℕ0s
112	111	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 1s ∈ ℕ0s)
113	58, 61, 112	pw2divscan4d 28338	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))))
114		exps1 28322	. . . . . . . . . . . . . . 15 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
115	2, 114	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2s↑s 1s ) = 2s
116	115	oveq1i 7359	. . . . . . . . . . . . 13 ⊢ ((2s↑s 1s ) ·s 𝐴) = (2s ·s 𝐴)
117	116	oveq1i 7359	. . . . . . . . . . . 12 ⊢ (((2s↑s 1s ) ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))) = ((2s ·s 𝐴) /su (2s↑s(𝑛 +s 1s )))
118	113, 117	eqtr2di 2781	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((2s ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))) = (𝐴 /su (2s↑s𝑛)))
119	109, 110, 118	3eqtr3d 2772	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) = (𝐴 /su (2s↑s𝑛)))
120	98, 119	breqtrd 5118	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (𝐴 /su (2s↑s𝑛)))
121	74, 86, 120	ssltsn 27704	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {(𝐴 /su (2s↑s𝑛))})
122	66, 64	addscomd 27881	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
123	122, 119	eqtrd 2764	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) = (𝐴 /su (2s↑s𝑛)))
124	88	simp2d 1143	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} <<s {({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})})
125		ovex 7382	. . . . . . . . . . . . . 14 ⊢ ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ V
126	125	snid 4614	. . . . . . . . . . . . 13 ⊢ ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}
127	126	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))})
128	124, 127, 92	ssltsepcd 27706	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) <s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
129	64, 73, 66	sltadd2d 27911	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) <s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ↔ (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
130	128, 129	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
131	123, 130	eqbrtrrd 5116	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 /su (2s↑s𝑛)) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
132	86, 75, 131	ssltsn 27704	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {(𝐴 /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
133	60, 61, 112	pw2divscan4d 28338	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s (𝐴 -s 1s )) /su (2s↑s(𝑛 +s 1s ))))
134	115	oveq1i 7359	. . . . . . . . . . . . . . . . 17 ⊢ ((2s↑s 1s ) ·s (𝐴 -s 1s )) = (2s ·s (𝐴 -s 1s ))
135		no2times 28311	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 -s 1s ) ∈ No → (2s ·s (𝐴 -s 1s )) = ((𝐴 -s 1s ) +s (𝐴 -s 1s )))
136	60, 135	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s (𝐴 -s 1s )) = ((𝐴 -s 1s ) +s (𝐴 -s 1s )))
137	134, 136	eqtrid 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((2s↑s 1s ) ·s (𝐴 -s 1s )) = ((𝐴 -s 1s ) +s (𝐴 -s 1s )))
138	137	oveq1d 7364	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((2s↑s 1s ) ·s (𝐴 -s 1s )) /su (2s↑s(𝑛 +s 1s ))) = (((𝐴 -s 1s ) +s (𝐴 -s 1s )) /su (2s↑s(𝑛 +s 1s ))))
139	60, 60, 63	pw2divsdird 28342	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) +s (𝐴 -s 1s )) /su (2s↑s(𝑛 +s 1s ))) = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))))
140	133, 138, 139	3eqtrrd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) = ((𝐴 -s 1s ) /su (2s↑s𝑛)))
141	64, 73, 64	sltadd2d 27911	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) <s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ↔ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
142	128, 141	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
143	140, 142	eqbrtrrd 5116	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 -s 1s ) /su (2s↑s𝑛)) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
144		sltasym 27658	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 -s 1s ) /su (2s↑s𝑛)) ∈ No ∧ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ∈ No ) → (((𝐴 -s 1s ) /su (2s↑s𝑛)) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) → ¬ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 -s 1s ) /su (2s↑s𝑛))))
145	79, 74, 144	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 -s 1s ) /su (2s↑s𝑛)) <s (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) → ¬ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 -s 1s ) /su (2s↑s𝑛))))
146	143, 145	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ¬ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 -s 1s ) /su (2s↑s𝑛)))
147	74, 79	ssltsnb 27703	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))} ↔ (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 -s 1s ) /su (2s↑s𝑛))))
148	146, 147	mtbird 325	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ¬ {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))})
149	148	intnanrd 489	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 -s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
150		ovex 7382	. . . . . . . . . . 11 ⊢ ((𝐴 -s 1s ) /su (2s↑s𝑛)) ∈ V
151		sneq 4587	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → {𝑥𝑂} = {((𝐴 -s 1s ) /su (2s↑s𝑛))})
152	151	breq2d 5104	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ↔ {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))}))
153	151	breq1d 5102	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → ({𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ↔ {((𝐴 -s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
154	152, 153	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → (({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 -s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})))
155	154	notbid 318	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → (¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 -s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})))
156	150, 155	ralsn 4633	. . . . . . . . . 10 ⊢ (∀𝑥𝑂 ∈ {((𝐴 -s 1s ) /su (2s↑s𝑛))} ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 -s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 -s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
157	149, 156	sylibr 234	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ∀𝑥𝑂 ∈ {((𝐴 -s 1s ) /su (2s↑s𝑛))} ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
158	73, 66, 66	sltadd2d 27911	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) <s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ↔ (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))))
159	96, 158	mpbid 232	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
160	65, 61, 112	pw2divscan4d 28338	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((𝐴 +s 1s ) /su (2s↑s𝑛)) = (((2s↑s 1s ) ·s (𝐴 +s 1s )) /su (2s↑s(𝑛 +s 1s ))))
161	115	oveq1i 7359	. . . . . . . . . . . . . . . . 17 ⊢ ((2s↑s 1s ) ·s (𝐴 +s 1s )) = (2s ·s (𝐴 +s 1s ))
162		no2times 28311	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +s 1s ) ∈ No → (2s ·s (𝐴 +s 1s )) = ((𝐴 +s 1s ) +s (𝐴 +s 1s )))
163	65, 162	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s (𝐴 +s 1s )) = ((𝐴 +s 1s ) +s (𝐴 +s 1s )))
164	161, 163	eqtrid 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((2s↑s 1s ) ·s (𝐴 +s 1s )) = ((𝐴 +s 1s ) +s (𝐴 +s 1s )))
165	164	oveq1d 7364	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((2s↑s 1s ) ·s (𝐴 +s 1s )) /su (2s↑s(𝑛 +s 1s ))) = (((𝐴 +s 1s ) +s (𝐴 +s 1s )) /su (2s↑s(𝑛 +s 1s ))))
166	65, 65, 63	pw2divsdird 28342	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) +s (𝐴 +s 1s )) /su (2s↑s(𝑛 +s 1s ))) = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
167	160, 165, 166	3eqtrrd 2769	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) = ((𝐴 +s 1s ) /su (2s↑s𝑛)))
168	159, 167	breqtrd 5118	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 +s 1s ) /su (2s↑s𝑛)))
169		sltasym 27658	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ∈ No ∧ ((𝐴 +s 1s ) /su (2s↑s𝑛)) ∈ No ) → ((((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 +s 1s ) /su (2s↑s𝑛)) → ¬ ((𝐴 +s 1s ) /su (2s↑s𝑛)) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
170	75, 80, 169	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) <s ((𝐴 +s 1s ) /su (2s↑s𝑛)) → ¬ ((𝐴 +s 1s ) /su (2s↑s𝑛)) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
171	168, 170	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ¬ ((𝐴 +s 1s ) /su (2s↑s𝑛)) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
172	80, 75	ssltsnb 27703	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ↔ ((𝐴 +s 1s ) /su (2s↑s𝑛)) <s (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
173	171, 172	mtbird 325	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ¬ {((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
174	173	intnand 488	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 +s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
175		ovex 7382	. . . . . . . . . . 11 ⊢ ((𝐴 +s 1s ) /su (2s↑s𝑛)) ∈ V
176		sneq 4587	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → {𝑥𝑂} = {((𝐴 +s 1s ) /su (2s↑s𝑛))})
177	176	breq2d 5104	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ↔ {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 +s 1s ) /su (2s↑s𝑛))}))
178	176	breq1d 5102	. . . . . . . . . . . . 13 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → ({𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ↔ {((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
179	177, 178	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → (({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 +s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})))
180	179	notbid 318	. . . . . . . . . . 11 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → (¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 +s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})))
181	175, 180	ralsn 4633	. . . . . . . . . 10 ⊢ (∀𝑥𝑂 ∈ {((𝐴 +s 1s ) /su (2s↑s𝑛))} ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {((𝐴 +s 1s ) /su (2s↑s𝑛))} ∧ {((𝐴 +s 1s ) /su (2s↑s𝑛))} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
182	174, 181	sylibr 234	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ∀𝑥𝑂 ∈ {((𝐴 +s 1s ) /su (2s↑s𝑛))} ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
183		ralunb 4148	. . . . . . . . 9 ⊢ (∀𝑥𝑂 ∈ ({((𝐴 -s 1s ) /su (2s↑s𝑛))} ∪ {((𝐴 +s 1s ) /su (2s↑s𝑛))}) ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ↔ (∀𝑥𝑂 ∈ {((𝐴 -s 1s ) /su (2s↑s𝑛))} ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) ∧ ∀𝑥𝑂 ∈ {((𝐴 +s 1s ) /su (2s↑s𝑛))} ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})))
184	157, 182, 183	sylanbrc 583	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ∀𝑥𝑂 ∈ ({((𝐴 -s 1s ) /su (2s↑s𝑛))} ∪ {((𝐴 +s 1s ) /su (2s↑s𝑛))}) ¬ ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} <<s {𝑥𝑂} ∧ {𝑥𝑂} <<s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
185	78, 83, 84, 85, 121, 132, 184	eqscut3 27736	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} |s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) = (𝐴 /su (2s↑s𝑛)))
186		no2times 28311	. . . . . . . . 9 ⊢ (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ No → (2s ·s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
187	73, 186	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
188		eqidd 2730	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
189	72, 72, 188, 188	addsunif 27916	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = (({𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)}) |s ({𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)})))
190		oveq1 7356	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
191	190	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ↔ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
192	125, 191	rexsn 4634	. . . . . . . . . . . . 13 ⊢ (∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ↔ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
193	192	abbii 2796	. . . . . . . . . . . 12 ⊢ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} = {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
194	193	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} = {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
195		oveq2 7357	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))))
196	195	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) ↔ 𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))))))
197	125, 196	rexsn 4634	. . . . . . . . . . . . 13 ⊢ (∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) ↔ 𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))))
198	73, 64	addscomd 27881	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
199	198	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))) ↔ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
200	197, 199	bitrid 283	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) ↔ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
201	200	abbidv 2795	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)} = {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
202	194, 201	uneq12d 4120	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)}) = ({𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
203		unidm 4108	. . . . . . . . . . 11 ⊢ ({𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) = {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
204		df-sn 4578	. . . . . . . . . . 11 ⊢ {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} = {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
205	203, 204	eqtr4i 2755	. . . . . . . . . 10 ⊢ ({𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ 𝑎 = (((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) = {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
206	202, 205	eqtrdi 2780	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)}) = {(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
207		oveq1 7356	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
208	207	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ↔ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
209	93, 208	rexsn 4634	. . . . . . . . . . . . 13 ⊢ (∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ↔ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
210	209	abbii 2796	. . . . . . . . . . . 12 ⊢ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} = {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
211	210	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} = {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
212		oveq2 7357	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
213	212	eqeq2d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑏 = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) → (𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) ↔ 𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))))
214	93, 213	rexsn 4634	. . . . . . . . . . . . 13 ⊢ (∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) ↔ 𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))))
215	73, 66	addscomd 27881	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
216	215	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))) ↔ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
217	214, 216	bitrid 283	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏) ↔ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
218	217	abbidv 2795	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)} = {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
219	211, 218	uneq12d 4120	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)}) = ({𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
220		unidm 4108	. . . . . . . . . . 11 ⊢ ({𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) = {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
221		df-sn 4578	. . . . . . . . . . 11 ⊢ {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} = {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
222	220, 221	eqtr4i 2755	. . . . . . . . . 10 ⊢ ({𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ 𝑎 = (((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}) = {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}
223	219, 222	eqtrdi 2780	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ({𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)}) = {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))})
224	206, 223	oveq12d 7367	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (({𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)}) |s ({𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (𝑏 +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} ∪ {𝑎 ∣ ∃𝑏 ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}𝑎 = (({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) +s 𝑏)})) = ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} |s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
225	187, 189, 224	3eqtrd 2768	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) = ({(((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))} |s {(((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) +s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))}))
226	2	a1i 11	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 2s ∈ No )
227	226, 58, 63	pw2divsassd 28337	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((2s ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))) = (2s ·s (𝐴 /su (2s↑s(𝑛 +s 1s )))))
228	117, 227	eqtr2id 2777	. . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s (𝐴 /su (2s↑s(𝑛 +s 1s )))) = (((2s↑s 1s ) ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))))
229	228, 113	eqtr4d 2767	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s (𝐴 /su (2s↑s(𝑛 +s 1s )))) = (𝐴 /su (2s↑s𝑛)))
230	185, 225, 229	3eqtr4rd 2775	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (2s ·s (𝐴 /su (2s↑s(𝑛 +s 1s )))) = (2s ·s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
231	58, 63	pw2divscld 28333	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 /su (2s↑s(𝑛 +s 1s ))) ∈ No )
232		2ne0s 28314	. . . . . . . 8 ⊢ 2s ≠ 0s
233	232	a1i 11	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 2s ≠ 0s )
234	231, 73, 226, 233	mulscan1d 28090	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → ((2s ·s (𝐴 /su (2s↑s(𝑛 +s 1s )))) = (2s ·s ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})) ↔ (𝐴 /su (2s↑s(𝑛 +s 1s ))) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})))
235	230, 234	mpbid 232	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 /su (2s↑s(𝑛 +s 1s ))) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))
236	235	3exp 1119	. . . 4 ⊢ (𝑛 ∈ ℕ0s → (𝐴 ∈ ℤs → ((𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))}) → (𝐴 /su (2s↑s(𝑛 +s 1s ))) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
237	236	a2d 29	. . 3 ⊢ (𝑛 ∈ ℕ0s → ((𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → (𝐴 ∈ ℤs → (𝐴 /su (2s↑s(𝑛 +s 1s ))) = ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}))))
238	13, 22, 31, 40, 56, 237	n0sind 28232	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))})))
239	238	impcom 407	1 ⊢ ((𝐴 ∈ ℤs ∧ 𝑁 ∈ ℕ0s) → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∪ cun 3901  {csn 4577   class class class wbr 5092  (class class class)co 7349   No csur 27549   <s cslt 27550   <<s csslt 27691   |s cscut 27693   0_s c0s 27737   1_s c1s 27738   +_s cadds 27873   -_s csubs 27933   ·_s cmuls 28016   /_su cdivs 28097  ℕ_0scnn0s 28213  ℤ_sczs 28273  2_sc2s 28304  ↑_scexps 28306

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 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-oadd 8392  df-nadd 8584  df-no 27552  df-slt 27553  df-bday 27554  df-sle 27655  df-sslt 27692  df-scut 27694  df-0s 27739  df-1s 27740  df-made 27759  df-old 27760  df-left 27762  df-right 27763  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-muls 28017  df-divs 28098  df-seqs 28185  df-n0s 28215  df-nns 28216  df-zs 28274  df-2s 28305  df-exps 28307

This theorem is referenced by: (None)