Theorem pw2cut2 28621

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 7419	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28578	. . . . . . . 8 ⊢ 2s ∈ No
3		exps0 28586	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2820	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq2d 7427	. . . . 5 ⊢ (𝑚 = 0s → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su 1s ))
7	5	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su 1s ))
8	7	sneqd 4604	. . . . . 6 ⊢ (𝑚 = 0s → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su 1s )})
9	5	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su 1s ))
10	9	sneqd 4604	. . . . . 6 ⊢ (𝑚 = 0s → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su 1s )})
11	8, 10	oveq12d 7429	. . . . 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 2785	. . . 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 343	. . 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 7419	. . . . . 6 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
15	14	oveq2d 7427	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s𝑛)))
16	14	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s𝑛)))
17	16	sneqd 4604	. . . . . 6 ⊢ (𝑚 = 𝑛 → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s𝑛))})
18	14	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s𝑛)))
19	18	sneqd 4604	. . . . . 6 ⊢ (𝑚 = 𝑛 → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s𝑛))})
20	17, 19	oveq12d 7429	. . . . 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 2785	. . . 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 343	. . 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 7419	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
24	23	oveq2d 7427	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s(𝑛 +s 1s ))))
25	23	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))))
26	25	sneqd 4604	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))})
27	23	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))
28	27	sneqd 4604	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
29	26, 28	oveq12d 7429	. . . . 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 2785	. . . 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 343	. . 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 7419	. . . . . 6 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
33	32	oveq2d 7427	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s𝑁)))
34	32	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s𝑁)))
35	34	sneqd 4604	. . . . . 6 ⊢ (𝑚 = 𝑁 → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s𝑁))})
36	32	oveq2d 7427	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s𝑁)))
37	36	sneqd 4604	. . . . . 6 ⊢ (𝑚 = 𝑁 → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s𝑁))})
38	35, 37	oveq12d 7429	. . . . 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 2785	. . . 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 343	. . 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		zcuts 28566	. . . 4 ⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
42		zno 28541	. . . . 5 ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
43	42	divs1d 28364	. . . 4 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = 𝐴)
44		1no 27969	. . . . . . . . 9 ⊢ 1s ∈ No
45	44	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℤs → 1s ∈ No )
46	42, 45	subscld 28222	. . . . . . 7 ⊢ (𝐴 ∈ ℤs → (𝐴 -s 1s ) ∈ No )
47	46	divs1d 28364	. . . . . 6 ⊢ (𝐴 ∈ ℤs → ((𝐴 -s 1s ) /su 1s ) = (𝐴 -s 1s ))
48	47	sneqd 4604	. . . . 5 ⊢ (𝐴 ∈ ℤs → {((𝐴 -s 1s ) /su 1s )} = {(𝐴 -s 1s )})
49	42, 45	addscld 28139	. . . . . . 7 ⊢ (𝐴 ∈ ℤs → (𝐴 +s 1s ) ∈ No )
50	49	divs1d 28364	. . . . . 6 ⊢ (𝐴 ∈ ℤs → ((𝐴 +s 1s ) /su 1s ) = (𝐴 +s 1s ))
51	50	sneqd 4604	. . . . 5 ⊢ (𝐴 ∈ ℤs → {((𝐴 +s 1s ) /su 1s )} = {(𝐴 +s 1s )})
52	48, 51	oveq12d 7429	. . . 4 ⊢ (𝐴 ∈ ℤs → ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
53	41, 43, 52	3eqtr4d 2814	. . 3 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}))
54		simp2 1153	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝐴 ∈ ℤs)
55	54	znod 28542	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝐴 ∈ No )
56	44	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 )
57	55, 56	subscld 28222	. . . . . . . . . . 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 )
58		simp1 1152	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ ℕ0s ∧ 𝐴 ∈ ℤs ∧ (𝐴 /su (2s↑s𝑛)) = ({((𝐴 -s 1s ) /su (2s↑s𝑛))} |s {((𝐴 +s 1s ) /su (2s↑s𝑛))})) → 𝑛 ∈ ℕ0s)
59		peano2n0s 28489	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
60	58, 59	syl 18	. . . . . . . . . . 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)
61	57, 60	pw2divscld 28598	. . . . . . . . . 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 )
62	55, 56	addscld 28139	. . . . . . . . . . . . 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 )
63	62, 60	pw2divscld 28598	. . . . . . . . . . . 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 )
64	55	ltsm1d 28261	. . . . . . . . . . . . . 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)
65	55	ltsp1d 28174	. . . . . . . . . . . . . 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 ))
66	57, 55, 62, 64, 65	ltstrd 27893	. . . . . . . . . . . . 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 ))
67	57, 62, 60	pw2ltsdiv1d 28611	. . . . . . . . . . . . 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 )))))
68	66, 67	mpbid 235	. . . . . . . . . . . 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 ))))
69	61, 63, 68	sltssn 27929	. . . . . . . . . . 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 )))})
70	69	cutscld 27942	. . . . . . . . . 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 )
71	61, 70	addscld 28139	. . . . . . . . 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 )
72	63, 70	addscld 28139	. . . . . . . . 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 )
73	61, 63, 70	ltadds1d 28157	. . . . . . . . . 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 )))}))))
74	68, 73	mpbid 235	. . . . . . . . 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 )))})))
75	71, 72, 74	sltssn 27929	. . . . . . . 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 )))}))})
76	57, 58	pw2divscld 28598	. . . . . . . . 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 )
77	62, 58	pw2divscld 28598	. . . . . . . . 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 )
78	57, 62, 58	pw2ltsdiv1d 28611	. . . . . . . . . 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))))
79	66, 78	mpbid 235	. . . . . . . . 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)))
80	76, 77, 79	sltssn 27929	. . . . . . . 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))})
81		eqidd 2770	. . . . . . . 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 )))}))}))
82		simp3 1154	. . . . . . . 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))}))
83	55, 58	pw2divscld 28598	. . . . . . . . 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 )
84		cutcuts 27940	. . . . . . . . . . . . . 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 )))}))
85	69, 84	syl 18	. . . . . . . . . . . . 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 )))}))
86	85	simp3d 1160	. . . . . . . . . . . 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 )))})
87		ovex 7444	. . . . . . . . . . . . . 14 ⊢ ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ V
88	87	snid 4631	. . . . . . . . . . . . 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 )))})}
89	88	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 )))})})
90		ovex 7444	. . . . . . . . . . . . . 14 ⊢ ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ V
91	90	snid 4631	. . . . . . . . . . . . 13 ⊢ ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ {((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 ))) ∈ {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
93	86, 89, 92	sltssepcd 27931	. . . . . . . . . . 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 ))))
94	70, 63, 61	ltadds2d 28156	. . . . . . . . . . 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 ))))))
95	93, 94	mpbid 235	. . . . . . . . . 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 )))))
96	55, 55, 56	addsassd 28165	. . . . . . . . . . . . . 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 )))
97	96	oveq1d 7426	. . . . . . . . . . . . 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 ))
98	55, 55	addscld 28139	. . . . . . . . . . . . . . 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 )
99		pncans 28231	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 +s 𝐴) ∈ No ∧ 1s ∈ No ) → (((𝐴 +s 𝐴) +s 1s ) -s 1s ) = (𝐴 +s 𝐴))
100	98, 44, 99	sylancl 597	. . . . . . . . . . . . . 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))
101		no2times 28576	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
102	55, 101	syl 18	. . . . . . . . . . . . . 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))
103	100, 102	eqtr4d 2807	. . . . . . . . . . . . 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))
104	55, 62, 56	addsubsd 28241	. . . . . . . . . . . . 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 )))
105	97, 103, 104	3eqtr3rd 2813	. . . . . . . . . . . 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))
106	105	oveq1d 7426	. . . . . . . . . . 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 ))))
107	57, 62, 60	pw2divsdird 28607	. . . . . . . . . . 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 )))))
108		1n0s 28507	. . . . . . . . . . . . . 14 ⊢ 1s ∈ ℕ0s
109	108	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)
110	55, 58, 109	pw2divscan4d 28603	. . . . . . . . . . . 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 ))))
111		exps1 28587	. . . . . . . . . . . . . . 15 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
112	2, 111	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2s↑s 1s ) = 2s
113	112	oveq1i 7421	. . . . . . . . . . . . 13 ⊢ ((2s↑s 1s ) ·s 𝐴) = (2s ·s 𝐴)
114	113	oveq1i 7421	. . . . . . . . . . . 12 ⊢ (((2s↑s 1s ) ·s 𝐴) /su (2s↑s(𝑛 +s 1s ))) = ((2s ·s 𝐴) /su (2s↑s(𝑛 +s 1s )))
115	110, 114	eqtr2di 2821	. . . . . . . . . . 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)))
116	106, 107, 115	3eqtr3d 2812	. . . . . . . . . 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)))
117	95, 116	breqtrd 5139	. . . . . . . . 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)))
118	71, 83, 117	sltssn 27929	. . . . . . . 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))})
119	63, 61	addscomd 28126	. . . . . . . . . . 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 )))))
120	119, 116	eqtrd 2804	. . . . . . . . . 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)))
121	85	simp2d 1159	. . . . . . . . . . . 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 )))})})
122		ovex 7444	. . . . . . . . . . . . . 14 ⊢ ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ V
123	122	snid 4631	. . . . . . . . . . . . 13 ⊢ ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))}
124	123	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 )))})
125	121, 124, 89	sltssepcd 27931	. . . . . . . . . . 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 )))}))
126	61, 70, 63	ltadds2d 28156	. . . . . . . . . . 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 )))}))))
127	125, 126	mpbid 235	. . . . . . . . . 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 )))})))
128	120, 127	eqbrtrrd 5137	. . . . . . . . 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 )))})))
129	83, 72, 128	sltssn 27929	. . . . . . . 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 )))}))})
130	57, 58, 109	pw2divscan4d 28603	. . . . . . . . . . . . . . 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 ))))
131	112	oveq1i 7421	. . . . . . . . . . . . . . . . 17 ⊢ ((2s↑s 1s ) ·s (𝐴 -s 1s )) = (2s ·s (𝐴 -s 1s ))
132		no2times 28576	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 -s 1s ) ∈ No → (2s ·s (𝐴 -s 1s )) = ((𝐴 -s 1s ) +s (𝐴 -s 1s )))
133	57, 132	syl 18	. . . . . . . . . . . . . . . . 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 )))
134	131, 133	eqtrid 2816	. . . . . . . . . . . . . . . 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 )))
135	134	oveq1d 7426	. . . . . . . . . . . . . . 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 ))))
136	57, 57, 60	pw2divsdird 28607	. . . . . . . . . . . . . . 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 )))))
137	130, 135, 136	3eqtrrd 2809	. . . . . . . . . . . . . 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)))
138	61, 70, 61	ltadds2d 28156	. . . . . . . . . . . . . . 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 )))}))))
139	125, 138	mpbid 235	. . . . . . . . . . . . . 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 )))})))
140	137, 139	eqbrtrrd 5137	. . . . . . . . . . . . 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 )))})))
141		ltsasym 27878	. . . . . . . . . . . . . 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))))
142	76, 71, 141	syl2anc 595	. . . . . . . . . . . . 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))))
143	140, 142	mpd 16	. . . . . . . . . . . 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)))
144	71, 76	sltssnb 27928	. . . . . . . . . . . 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))))
145	143, 144	mtbird 328	. . . . . . . . . . 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))})
146	145	intnanrd 494	. . . . . . . . . 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 )))}))}))
147		ovex 7444	. . . . . . . . . . 11 ⊢ ((𝐴 -s 1s ) /su (2s↑s𝑛)) ∈ V
148		sneq 4602	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → {𝑥𝑂} = {((𝐴 -s 1s ) /su (2s↑s𝑛))})
149	148	breq2d 5123	. . . . . . . . . . . . 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))}))
150	148	breq1d 5121	. . . . . . . . . . . . 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 )))}))}))
151	149, 150	anbi12d 643	. . . . . . . . . . . 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 )))}))})))
152	151	notbid 321	. . . . . . . . . . 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 )))}))})))
153	147, 152	ralsn 4650	. . . . . . . . . 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 )))}))}))
154	146, 153	sylibr 237	. . . . . . . . 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 )))}))}))
155	70, 63, 63	ltadds2d 28156	. . . . . . . . . . . . . . 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 ))))))
156	93, 155	mpbid 235	. . . . . . . . . . . . . 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 )))))
157	62, 58, 109	pw2divscan4d 28603	. . . . . . . . . . . . . . 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 ))))
158	112	oveq1i 7421	. . . . . . . . . . . . . . . . 17 ⊢ ((2s↑s 1s ) ·s (𝐴 +s 1s )) = (2s ·s (𝐴 +s 1s ))
159		no2times 28576	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +s 1s ) ∈ No → (2s ·s (𝐴 +s 1s )) = ((𝐴 +s 1s ) +s (𝐴 +s 1s )))
160	62, 159	syl 18	. . . . . . . . . . . . . . . . 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 )))
161	158, 160	eqtrid 2816	. . . . . . . . . . . . . . . 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 )))
162	161	oveq1d 7426	. . . . . . . . . . . . . . 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 ))))
163	62, 62, 60	pw2divsdird 28607	. . . . . . . . . . . . . . 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 )))))
164	157, 162, 163	3eqtrrd 2809	. . . . . . . . . . . . . 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)))
165	156, 164	breqtrd 5139	. . . . . . . . . . . . 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)))
166		ltsasym 27878	. . . . . . . . . . . . . 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 )))}))))
167	72, 77, 166	syl2anc 595	. . . . . . . . . . . . 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 )))}))))
168	165, 167	mpd 16	. . . . . . . . . . . 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 )))})))
169	77, 72	sltssnb 27928	. . . . . . . . . . . 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 )))}))))
170	168, 169	mtbird 328	. . . . . . . . . . 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 )))}))})
171	170	intnand 493	. . . . . . . . . 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 )))}))}))
172		ovex 7444	. . . . . . . . . . 11 ⊢ ((𝐴 +s 1s ) /su (2s↑s𝑛)) ∈ V
173		sneq 4602	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → {𝑥𝑂} = {((𝐴 +s 1s ) /su (2s↑s𝑛))})
174	173	breq2d 5123	. . . . . . . . . . . . 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))}))
175	173	breq1d 5121	. . . . . . . . . . . . 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 )))}))}))
176	174, 175	anbi12d 643	. . . . . . . . . . . 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 )))}))})))
177	176	notbid 321	. . . . . . . . . . 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 )))}))})))
178	172, 177	ralsn 4650	. . . . . . . . . 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 )))}))}))
179	171, 178	sylibr 237	. . . . . . . . 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 )))}))}))
180		ralunb 4156	. . . . . . . . 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 )))}))})))
181	154, 179, 180	sylanbrc 594	. . . . . . . 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 )))}))}))
182	75, 80, 81, 82, 118, 129, 181	eqcuts3 27963	. . . . . . 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)))
183		no2times 28576	. . . . . . . . 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 )))})))
184	70, 183	syl 18	. . . . . . . 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 )))})))
185		eqidd 2770	. . . . . . . . 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 )))}))
186	69, 69, 185, 185	addsunif 28161	. . . . . . . 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)})))
187		oveq1 7418	. . . . . . . . . . . . . . 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 )))})))
188	187	eqeq2d 2780	. . . . . . . . . . . . . 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 )))}))))
189	122, 188	rexsn 4651	. . . . . . . . . . . . 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 )))})))
190	189	abbii 2836	. . . . . . . . . . . 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 )))}))}
191	190	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 )))}))})
192		oveq2 7419	. . . . . . . . . . . . . . 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 )))))
193	192	eqeq2d 2780	. . . . . . . . . . . . . 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 ))))))
194	122, 193	rexsn 4651	. . . . . . . . . . . . 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 )))))
195	70, 61	addscomd 28126	. . . . . . . . . . . . . 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 )))})))
196	195	eqeq2d 2780	. . . . . . . . . . . . 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 )))}))))
197	194, 196	bitrid 286	. . . . . . . . . . . 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 )))}))))
198	197	abbidv 2835	. . . . . . . . . . 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 )))}))})
199	191, 198	uneq12d 4129	. . . . . . . . . 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 )))}))}))
200		unidm 4117	. . . . . . . . . . 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 )))}))}
201		df-sn 4593	. . . . . . . . . . 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 )))}))}
202	200, 201	eqtr4i 2795	. . . . . . . . . 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 )))}))}
203	199, 202	eqtrdi 2820	. . . . . . . . 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 )))}))})
204		oveq1 7418	. . . . . . . . . . . . . . 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 )))})))
205	204	eqeq2d 2780	. . . . . . . . . . . . . 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 )))}))))
206	90, 205	rexsn 4651	. . . . . . . . . . . . 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 )))})))
207	206	abbii 2836	. . . . . . . . . . . 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 )))}))}
208	207	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 )))}))})
209		oveq2 7419	. . . . . . . . . . . . . . 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 )))))
210	209	eqeq2d 2780	. . . . . . . . . . . . . 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 ))))))
211	90, 210	rexsn 4651	. . . . . . . . . . . . 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 )))))
212	70, 63	addscomd 28126	. . . . . . . . . . . . . 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 )))})))
213	212	eqeq2d 2780	. . . . . . . . . . . . 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 )))}))))
214	211, 213	bitrid 286	. . . . . . . . . . . 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 )))}))))
215	214	abbidv 2835	. . . . . . . . . . 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 )))}))})
216	208, 215	uneq12d 4129	. . . . . . . . . 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 )))}))}))
217		unidm 4117	. . . . . . . . . . 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 )))}))}
218		df-sn 4593	. . . . . . . . . . 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 )))}))}
219	217, 218	eqtr4i 2795	. . . . . . . . . 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 )))}))}
220	216, 219	eqtrdi 2820	. . . . . . . . 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 )))}))})
221	203, 220	oveq12d 7429	. . . . . . . 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 )))}))}))
222	184, 186, 221	3eqtrd 2808	. . . . . . 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 )))}))}))
223	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 )
224	223, 55, 60	pw2divsassd 28602	. . . . . . . . 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 )))))
225	114, 224	eqtr2id 2817	. . . . . . . 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 ))))
226	225, 110	eqtr4d 2807	. . . . . . 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)))
227	182, 222, 226	3eqtr4rd 2815	. . . . . 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 )))})))
228	55, 60	pw2divscld 28598	. . . . . . 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 )
229		2ne0s 28579	. . . . . . . 8 ⊢ 2s ≠ 0s
230	229	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 )
231	228, 70, 223, 230	mulscan1d 28339	. . . . . 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 )))})))
232	227, 231	mpbid 235	. . . . 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 )))}))
233	232	3exp 1135	. . . 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 )))}))))
234	233	a2d 30	. . 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 )))}))))
235	13, 22, 31, 40, 53, 234	n0sind 28492	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝐴 ∈ ℤs → (𝐴 /su (2s↑s𝑁)) = ({((𝐴 -s 1s ) /su (2s↑s𝑁))} |s {((𝐴 +s 1s ) /su (2s↑s𝑁))})))
236	235	impcom 412	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 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  {cab 2747   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∪ cun 3909  {csn 4592   class class class wbr 5111  (class class class)co 7411   No csur 27770   <s clts 27771   <<s cslts 27916   |s ccuts 27918   0_s c0s 27964   1_s c1s 27965   +_s cadds 28118   -_s csubs 28179   ·_s cmuls 28265   /_su cdivs 28346  ℕ_0scn0s 28471  ℤ_sczs 28537  2_sc2s 28569  ↑_scexps 28571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-nadd 8652  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967  df-made 27986  df-old 27987  df-left 27989  df-right 27990  df-norec 28097  df-norec2 28108  df-adds 28119  df-negs 28180  df-subs 28181  df-muls 28266  df-divs 28347  df-seqs 28443  df-n0s 28473  df-nns 28474  df-zs 28538  df-2s 28570  df-exps 28572

This theorem is referenced by: (None)