Theorem pw2cut2 28462

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 7368	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
2		2no 28419	. . . . . . . 8 ⊢ 2s ∈ No
3		exps0 28427	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 0s ) = 1s
5	1, 4	eqtrdi 2788	. . . . . 6 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
6	5	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 0s → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su 1s ))
7	5	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su 1s ))
8	7	sneqd 4593	. . . . . 6 ⊢ (𝑚 = 0s → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su 1s )})
9	5	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su 1s ))
10	9	sneqd 4593	. . . . . 6 ⊢ (𝑚 = 0s → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su 1s )})
11	8, 10	oveq12d 7378	. . . . 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 2753	. . . 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 7368	. . . . . 6 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
15	14	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s𝑛)))
16	14	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s𝑛)))
17	16	sneqd 4593	. . . . . 6 ⊢ (𝑚 = 𝑛 → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s𝑛))})
18	14	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s𝑛)))
19	18	sneqd 4593	. . . . . 6 ⊢ (𝑚 = 𝑛 → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s𝑛))})
20	17, 19	oveq12d 7378	. . . . 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 2753	. . . 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 7368	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
24	23	oveq2d 7376	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s(𝑛 +s 1s ))))
25	23	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))))
26	25	sneqd 4593	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))})
27	23	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))))
28	27	sneqd 4593	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))})
29	26, 28	oveq12d 7378	. . . . 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 2753	. . . 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 7368	. . . . . 6 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
33	32	oveq2d 7376	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝐴 /su (2s↑s𝑚)) = (𝐴 /su (2s↑s𝑁)))
34	32	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝐴 -s 1s ) /su (2s↑s𝑚)) = ((𝐴 -s 1s ) /su (2s↑s𝑁)))
35	34	sneqd 4593	. . . . . 6 ⊢ (𝑚 = 𝑁 → {((𝐴 -s 1s ) /su (2s↑s𝑚))} = {((𝐴 -s 1s ) /su (2s↑s𝑁))})
36	32	oveq2d 7376	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝐴 +s 1s ) /su (2s↑s𝑚)) = ((𝐴 +s 1s ) /su (2s↑s𝑁)))
37	36	sneqd 4593	. . . . . 6 ⊢ (𝑚 = 𝑁 → {((𝐴 +s 1s ) /su (2s↑s𝑚))} = {((𝐴 +s 1s ) /su (2s↑s𝑁))})
38	35, 37	oveq12d 7378	. . . . 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 2753	. . . 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		zcuts 28407	. . . 4 ⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
42		zno 28382	. . . . 5 ⊢ (𝐴 ∈ ℤs → 𝐴 ∈ No )
43	42	divs1d 28205	. . . 4 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = 𝐴)
44		1no 27810	. . . . . . . . 9 ⊢ 1s ∈ No
45	44	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℤs → 1s ∈ No )
46	42, 45	subscld 28063	. . . . . . 7 ⊢ (𝐴 ∈ ℤs → (𝐴 -s 1s ) ∈ No )
47	46	divs1d 28205	. . . . . 6 ⊢ (𝐴 ∈ ℤs → ((𝐴 -s 1s ) /su 1s ) = (𝐴 -s 1s ))
48	47	sneqd 4593	. . . . 5 ⊢ (𝐴 ∈ ℤs → {((𝐴 -s 1s ) /su 1s )} = {(𝐴 -s 1s )})
49	42, 45	addscld 27980	. . . . . . 7 ⊢ (𝐴 ∈ ℤs → (𝐴 +s 1s ) ∈ No )
50	49	divs1d 28205	. . . . . 6 ⊢ (𝐴 ∈ ℤs → ((𝐴 +s 1s ) /su 1s ) = (𝐴 +s 1s ))
51	50	sneqd 4593	. . . . 5 ⊢ (𝐴 ∈ ℤs → {((𝐴 +s 1s ) /su 1s )} = {(𝐴 +s 1s )})
52	48, 51	oveq12d 7378	. . . 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 2782	. . 3 ⊢ (𝐴 ∈ ℤs → (𝐴 /su 1s ) = ({((𝐴 -s 1s ) /su 1s )} |s {((𝐴 +s 1s ) /su 1s )}))
54		simp2 1138	. . . . . . . . . . . . 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 28383	. . . . . . . . . . . 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 28063	. . . . . . . . . . 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 1137	. . . . . . . . . . . 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 28330	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
60	58, 59	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)
61	57, 60	pw2divscld 28439	. . . . . . . . . 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 27980	. . . . . . . . . . . . 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 28439	. . . . . . . . . . . 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 28102	. . . . . . . . . . . . . 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 28015	. . . . . . . . . . . . . 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 27735	. . . . . . . . . . . . 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 28452	. . . . . . . . . . . . 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 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 ))))
69	61, 63, 68	sltssn 27770	. . . . . . . . . . 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 27783	. . . . . . . . . 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 27980	. . . . . . . . 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 27980	. . . . . . . . 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 27998	. . . . . . . . . 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 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 )))})))
75	71, 72, 74	sltssn 27770	. . . . . . . 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 28439	. . . . . . . . 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 28439	. . . . . . . . 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 28452	. . . . . . . . . 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 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)))
80	76, 77, 79	sltssn 27770	. . . . . . . 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 2738	. . . . . . . 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 1139	. . . . . . . 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 28439	. . . . . . . . 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 27781	. . . . . . . . . . . . . 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 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 )))}))
86	85	simp3d 1145	. . . . . . . . . . . 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 7393	. . . . . . . . . . . . . 14 ⊢ ({((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s )))} |s {((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s )))}) ∈ V
88	87	snid 4620	. . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . 14 ⊢ ((𝐴 +s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ V
91	90	snid 4620	. . . . . . . . . . . . 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 27772	. . . . . . . . . . 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 27997	. . . . . . . . . . 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 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 )))))
96	55, 55, 56	addsassd 28006	. . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . 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 27980	. . . . . . . . . . . . . . 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 28072	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 +s 𝐴) ∈ No ∧ 1s ∈ No ) → (((𝐴 +s 𝐴) +s 1s ) -s 1s ) = (𝐴 +s 𝐴))
100	98, 44, 99	sylancl 587	. . . . . . . . . . . . . 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 28417	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ No → (2s ·s 𝐴) = (𝐴 +s 𝐴))
102	55, 101	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))
103	100, 102	eqtr4d 2775	. . . . . . . . . . . . 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 28082	. . . . . . . . . . . . 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 2781	. . . . . . . . . . . 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 7375	. . . . . . . . . . 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 28448	. . . . . . . . . . 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 28348	. . . . . . . . . . . . . 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 28444	. . . . . . . . . . . 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 28428	. . . . . . . . . . . . . . 15 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
112	2, 111	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (2s↑s 1s ) = 2s
113	112	oveq1i 7370	. . . . . . . . . . . . 13 ⊢ ((2s↑s 1s ) ·s 𝐴) = (2s ·s 𝐴)
114	113	oveq1i 7370	. . . . . . . . . . . 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 2789	. . . . . . . . . . 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 2780	. . . . . . . . . 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 5125	. . . . . . . . 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 27770	. . . . . . . 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 27967	. . . . . . . . . . 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 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)))
121	85	simp2d 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 )))})})
122		ovex 7393	. . . . . . . . . . . . . 14 ⊢ ((𝐴 -s 1s ) /su (2s↑s(𝑛 +s 1s ))) ∈ V
123	122	snid 4620	. . . . . . . . . . . . 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 27772	. . . . . . . . . . 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 27997	. . . . . . . . . . 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 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 )))})))
128	120, 127	eqbrtrrd 5123	. . . . . . . . 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 27770	. . . . . . . 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 28444	. . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . . 17 ⊢ ((2s↑s 1s ) ·s (𝐴 -s 1s )) = (2s ·s (𝐴 -s 1s ))
132		no2times 28417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 -s 1s ) ∈ No → (2s ·s (𝐴 -s 1s )) = ((𝐴 -s 1s ) +s (𝐴 -s 1s )))
133	57, 132	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 )))
134	131, 133	eqtrid 2784	. . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . 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 28448	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 27997	. . . . . . . . . . . . . . 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 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 )))})))
140	137, 139	eqbrtrrd 5123	. . . . . . . . . . . . 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 27720	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 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)))
144	71, 76	sltssnb 27769	. . . . . . . . . . . 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 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))})
146	145	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 )))}))}))
147		ovex 7393	. . . . . . . . . . 11 ⊢ ((𝐴 -s 1s ) /su (2s↑s𝑛)) ∈ V
148		sneq 4591	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = ((𝐴 -s 1s ) /su (2s↑s𝑛)) → {𝑥𝑂} = {((𝐴 -s 1s ) /su (2s↑s𝑛))})
149	148	breq2d 5111	. . . . . . . . . . . . 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 5109	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 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 )))}))})))
153	147, 152	ralsn 4639	. . . . . . . . . 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 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 )))}))}))
155	70, 63, 63	ltadds2d 27997	. . . . . . . . . . . . . . 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 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 )))))
157	62, 58, 109	pw2divscan4d 28444	. . . . . . . . . . . . . . 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 7370	. . . . . . . . . . . . . . . . 17 ⊢ ((2s↑s 1s ) ·s (𝐴 +s 1s )) = (2s ·s (𝐴 +s 1s ))
159		no2times 28417	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 +s 1s ) ∈ No → (2s ·s (𝐴 +s 1s )) = ((𝐴 +s 1s ) +s (𝐴 +s 1s )))
160	62, 159	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 )))
161	158, 160	eqtrid 2784	. . . . . . . . . . . . . . . 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 7375	. . . . . . . . . . . . . . 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 28448	. . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . 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 5125	. . . . . . . . . . . . 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 27720	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 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 )))})))
169	77, 72	sltssnb 27769	. . . . . . . . . . . 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 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 )))}))})
171	170	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 )))}))}))
172		ovex 7393	. . . . . . . . . . 11 ⊢ ((𝐴 +s 1s ) /su (2s↑s𝑛)) ∈ V
173		sneq 4591	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑂 = ((𝐴 +s 1s ) /su (2s↑s𝑛)) → {𝑥𝑂} = {((𝐴 +s 1s ) /su (2s↑s𝑛))})
174	173	breq2d 5111	. . . . . . . . . . . . 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 5109	. . . . . . . . . . . . 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 633	. . . . . . . . . . . 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 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 )))}))})))
178	172, 177	ralsn 4639	. . . . . . . . . 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 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 )))}))}))
180		ralunb 4150	. . . . . . . . 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 584	. . . . . . . 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 27804	. . . . . . 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 28417	. . . . . . . . 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 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 )))})))
185		eqidd 2738	. . . . . . . . 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 28002	. . . . . . . 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 7367	. . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . 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 4640	. . . . . . . . . . . . 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 2804	. . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . 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 4640	. . . . . . . . . . . . 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 27967	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . 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 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 )))}))))
198	197	abbidv 2803	. . . . . . . . . . 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 4122	. . . . . . . . . 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 4110	. . . . . . . . . . 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 4582	. . . . . . . . . . 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 2763	. . . . . . . . . 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 2788	. . . . . . . . 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 7367	. . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . 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 4640	. . . . . . . . . . . . 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 2804	. . . . . . . . . . . 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 7368	. . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . 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 4640	. . . . . . . . . . . . 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 27967	. . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . 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 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 )))}))))
215	214	abbidv 2803	. . . . . . . . . . 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 4122	. . . . . . . . . 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 4110	. . . . . . . . . . 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 4582	. . . . . . . . . . 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 2763	. . . . . . . . . 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 2788	. . . . . . . . 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 7378	. . . . . . . 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 2776	. . . . . . 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 28443	. . . . . . . . 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 2785	. . . . . . . 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 2775	. . . . . . 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 2783	. . . . . 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 28439	. . . . . . 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 28420	. . . . . . . 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 28180	. . . . . 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 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 )))}))
233	232	3exp 1120	. . . 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 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 )))}))))
235	13, 22, 31, 40, 53, 234	n0sind 28333	. 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 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 1087   = wceq 1542   ∈ wcel 2114  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3061   ∪ cun 3900  {csn 4581   class class class wbr 5099  (class class class)co 7360   No csur 27611   <s clts 27612   <<s cslts 27757   |s ccuts 27759   0_s c0s 27805   1_s c1s 27806   +_s cadds 27959   -_s csubs 28020   ·_s cmuls 28106   /_su cdivs 28187  ℕ_0scn0s 28312  ℤ_sczs 28378  2_sc2s 28410  ↑_scexps 28412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-ot 4590  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-nadd 8596  df-no 27614  df-lts 27615  df-bday 27616  df-les 27717  df-slts 27758  df-cuts 27760  df-0s 27807  df-1s 27808  df-made 27827  df-old 27828  df-left 27830  df-right 27831  df-norec 27938  df-norec2 27949  df-adds 27960  df-negs 28021  df-subs 28022  df-muls 28107  df-divs 28188  df-seqs 28284  df-n0s 28314  df-nns 28315  df-zs 28379  df-2s 28411  df-exps 28413

This theorem is referenced by: (None)