Theorem pw2cut 28519

Description: Extend halfcut 28517 to arbitrary powers of two. Part of theorem 4.2 of [Gonshor] p. 28. (Contributed by Scott Fenton, 18-Aug-2025.)

Hypotheses

Ref	Expression
pw2cut.1	⊢ (𝜑 → 𝐴 ∈ No )
pw2cut.2	⊢ (𝜑 → 𝐵 ∈ No )
pw2cut.3	⊢ (𝜑 → 𝑁 ∈ ℕ0s)
pw2cut.4	⊢ (𝜑 → 𝐴 <s 𝐵)
pw2cut.5	⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))

Assertion

Ref	Expression
pw2cut	⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s ))))

Proof of Theorem pw2cut

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw2cut.3	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0s)
2		oveq2 7389	. . . . . . . . 9 ⊢ (𝑥 = 0s → (2s↑s𝑥) = (2s↑s 0s ))
3		2no 28478	. . . . . . . . . 10 ⊢ 2s ∈ No
4		exps0 28486	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
5	3, 4	ax-mp 5	. . . . . . . . 9 ⊢ (2s↑s 0s ) = 1s
6	2, 5	eqtrdi 2803	. . . . . . . 8 ⊢ (𝑥 = 0s → (2s↑s𝑥) = 1s )
7	6	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 0s → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su 1s ))
8	7	sneqd 4584	. . . . . 6 ⊢ (𝑥 = 0s → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su 1s )})
9	6	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 0s → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su 1s ))
10	9	sneqd 4584	. . . . . 6 ⊢ (𝑥 = 0s → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su 1s )})
11	8, 10	oveq12d 7399	. . . . 5 ⊢ (𝑥 = 0s → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}))
12		oveq1 7388	. . . . . . . . 9 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = ( 0s +s 1s ))
13		1no 27869	. . . . . . . . . 10 ⊢ 1s ∈ No
14		addslid 28027	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
16	12, 15	eqtrdi 2803	. . . . . . . 8 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = 1s )
17	16	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 0s → (2s↑s(𝑥 +s 1s )) = (2s↑s 1s ))
18		exps1 28487	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
19	3, 18	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
20	17, 19	eqtrdi 2803	. . . . . 6 ⊢ (𝑥 = 0s → (2s↑s(𝑥 +s 1s )) = 2s)
21	20	oveq2d 7397	. . . . 5 ⊢ (𝑥 = 0s → ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) = ((𝐴 +s 𝐵) /su 2s))
22	11, 21	eqeq12d 2768	. . . 4 ⊢ (𝑥 = 0s → (({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) ↔ ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}) = ((𝐴 +s 𝐵) /su 2s)))
23	22	imbi2d 342	. . 3 ⊢ (𝑥 = 0s → ((𝜑 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s )))) ↔ (𝜑 → ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}) = ((𝐴 +s 𝐵) /su 2s))))
24		oveq2 7389	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2s↑s𝑥) = (2s↑s𝑦))
25	24	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su (2s↑s𝑦)))
26	25	sneqd 4584	. . . . . 6 ⊢ (𝑥 = 𝑦 → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su (2s↑s𝑦))})
27	24	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su (2s↑s𝑦)))
28	27	sneqd 4584	. . . . . 6 ⊢ (𝑥 = 𝑦 → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su (2s↑s𝑦))})
29	26, 28	oveq12d 7399	. . . . 5 ⊢ (𝑥 = 𝑦 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}))
30		oveq1 7388	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
31	30	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2s↑s(𝑥 +s 1s )) = (2s↑s(𝑦 +s 1s )))
32	31	oveq2d 7397	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))))
33	29, 32	eqeq12d 2768	. . . 4 ⊢ (𝑥 = 𝑦 → (({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) ↔ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))))
34	33	imbi2d 342	. . 3 ⊢ (𝑥 = 𝑦 → ((𝜑 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s )))) ↔ (𝜑 → ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))))))
35		oveq2 7389	. . . . . . . 8 ⊢ (𝑥 = (𝑦 +s 1s ) → (2s↑s𝑥) = (2s↑s(𝑦 +s 1s )))
36	35	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su (2s↑s(𝑦 +s 1s ))))
37	36	sneqd 4584	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su (2s↑s(𝑦 +s 1s )))})
38	35	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su (2s↑s(𝑦 +s 1s ))))
39	38	sneqd 4584	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su (2s↑s(𝑦 +s 1s )))})
40	37, 39	oveq12d 7399	. . . . 5 ⊢ (𝑥 = (𝑦 +s 1s ) → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}))
41		oveq1 7388	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 +s 1s ) = ((𝑦 +s 1s ) +s 1s ))
42	41	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (2s↑s(𝑥 +s 1s )) = (2s↑s((𝑦 +s 1s ) +s 1s )))
43	42	oveq2d 7397	. . . . 5 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) = ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))))
44	40, 43	eqeq12d 2768	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) ↔ ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s )))))
45	44	imbi2d 342	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝜑 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s )))) ↔ (𝜑 → ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))))))
46		oveq2 7389	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (2s↑s𝑥) = (2s↑s𝑁))
47	46	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su (2s↑s𝑁)))
48	47	sneqd 4584	. . . . . 6 ⊢ (𝑥 = 𝑁 → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su (2s↑s𝑁))})
49	46	oveq2d 7397	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su (2s↑s𝑁)))
50	49	sneqd 4584	. . . . . 6 ⊢ (𝑥 = 𝑁 → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su (2s↑s𝑁))})
51	48, 50	oveq12d 7399	. . . . 5 ⊢ (𝑥 = 𝑁 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}))
52		oveq1 7388	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 +s 1s ) = (𝑁 +s 1s ))
53	52	oveq2d 7397	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2s↑s(𝑥 +s 1s )) = (2s↑s(𝑁 +s 1s )))
54	53	oveq2d 7397	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s ))))
55	51, 54	eqeq12d 2768	. . . 4 ⊢ (𝑥 = 𝑁 → (({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) ↔ ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s )))))
56	55	imbi2d 342	. . 3 ⊢ (𝑥 = 𝑁 → ((𝜑 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s )))) ↔ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s ))))))
57		pw2cut.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ No )
58	57	divs1d 28264	. . . . . 6 ⊢ (𝜑 → (𝐴 /su 1s ) = 𝐴)
59	58	sneqd 4584	. . . . 5 ⊢ (𝜑 → {(𝐴 /su 1s )} = {𝐴})
60		pw2cut.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
61	60	divs1d 28264	. . . . . 6 ⊢ (𝜑 → (𝐵 /su 1s ) = 𝐵)
62	61	sneqd 4584	. . . . 5 ⊢ (𝜑 → {(𝐵 /su 1s )} = {𝐵})
63	59, 62	oveq12d 7399	. . . 4 ⊢ (𝜑 → ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}) = ({𝐴} |s {𝐵}))
64		pw2cut.4	. . . . 5 ⊢ (𝜑 → 𝐴 <s 𝐵)
65		pw2cut.5	. . . . 5 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))
66		eqid 2752	. . . . 5 ⊢ ({𝐴} |s {𝐵}) = ({𝐴} |s {𝐵})
67	57, 60, 64, 65, 66	halfcut 28517	. . . 4 ⊢ (𝜑 → ({𝐴} |s {𝐵}) = ((𝐴 +s 𝐵) /su 2s))
68	63, 67	eqtrd 2787	. . 3 ⊢ (𝜑 → ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}) = ((𝐴 +s 𝐵) /su 2s))
69	57	adantl 484	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝐴 ∈ No )
70		peano2n0s 28389	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
71		expscl 28490	. . . . . . . . . . . 12 ⊢ ((2s ∈ No ∧ (𝑦 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑦 +s 1s )) ∈ No )
72	3, 70, 71	sylancr 595	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) ∈ No )
73	72	adantr 483	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s(𝑦 +s 1s )) ∈ No )
74		2ne0s 28479	. . . . . . . . . . . . 13 ⊢ 2s ≠ 0s
75		expsne0 28495	. . . . . . . . . . . . 13 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ (𝑦 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑦 +s 1s )) ≠ 0s )
76	3, 74, 75	mp3an12 1462	. . . . . . . . . . . 12 ⊢ ((𝑦 +s 1s ) ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) ≠ 0s )
77	70, 76	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) ≠ 0s )
78	77	adantr 483	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s(𝑦 +s 1s )) ≠ 0s )
79	69, 73, 78	divscld 28283	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) ∈ No )
80	79	3adant3 1141	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) ∈ No )
81	60	adantl 484	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝐵 ∈ No )
82	81, 73, 78	divscld 28283	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) ∈ No )
83	82	3adant3 1141	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) ∈ No )
84	69, 73, 78	divscan1d 28285	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 /su (2s↑s(𝑦 +s 1s ))) ·s (2s↑s(𝑦 +s 1s ))) = 𝐴)
85	64	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝐴 <s 𝐵)
86	84, 85	eqbrtrd 5112	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 /su (2s↑s(𝑦 +s 1s ))) ·s (2s↑s(𝑦 +s 1s ))) <s 𝐵)
87		2nns 28477	. . . . . . . . . . . . . . 15 ⊢ 2s ∈ ℕs
88		nnsgt0 28398	. . . . . . . . . . . . . . 15 ⊢ (2s ∈ ℕs → 0s <s 2s)
89	87, 88	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 0s <s 2s
90		expsgt0 28496	. . . . . . . . . . . . . 14 ⊢ ((2s ∈ No ∧ (𝑦 +s 1s ) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑦 +s 1s )))
91	3, 89, 90	mp3an13 1463	. . . . . . . . . . . . 13 ⊢ ((𝑦 +s 1s ) ∈ ℕ0s → 0s <s (2s↑s(𝑦 +s 1s )))
92	70, 91	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 0s <s (2s↑s(𝑦 +s 1s )))
93	92	adantr 483	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 0s <s (2s↑s(𝑦 +s 1s )))
94	79, 81, 73, 93	ltmuldivsd 28288	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (((𝐴 /su (2s↑s(𝑦 +s 1s ))) ·s (2s↑s(𝑦 +s 1s ))) <s 𝐵 ↔ (𝐴 /su (2s↑s(𝑦 +s 1s ))) <s (𝐵 /su (2s↑s(𝑦 +s 1s )))))
95	86, 94	mpbid 234	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) <s (𝐵 /su (2s↑s(𝑦 +s 1s ))))
96	95	3adant3 1141	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) <s (𝐵 /su (2s↑s(𝑦 +s 1s ))))
97		expsp1 28488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑦 ∈ ℕ0s) → (2s↑s(𝑦 +s 1s )) = ((2s↑s𝑦) ·s 2s))
98	3, 97	mpan 698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) = ((2s↑s𝑦) ·s 2s))
99	98	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s(𝑦 +s 1s )) = ((2s↑s𝑦) ·s 2s))
100	99	oveq2d 7397	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) = (𝐴 /su ((2s↑s𝑦) ·s 2s)))
101		expscl 28490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑦 ∈ ℕ0s) → (2s↑s𝑦) ∈ No )
102	3, 101	mpan 698	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0s → (2s↑s𝑦) ∈ No )
103	102	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s𝑦) ∈ No )
104	3	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 2s ∈ No )
105		expsne0 28495	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ 𝑦 ∈ ℕ0s) → (2s↑s𝑦) ≠ 0s )
106	3, 74, 105	mp3an12 1462	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0s → (2s↑s𝑦) ≠ 0s )
107	106	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s𝑦) ≠ 0s )
108	74	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 2s ≠ 0s )
109	69, 103, 104, 107, 108	divdivs1d 28292	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 /su (2s↑s𝑦)) /su 2s) = (𝐴 /su ((2s↑s𝑦) ·s 2s)))
110	100, 109	eqtr4d 2790	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) = ((𝐴 /su (2s↑s𝑦)) /su 2s))
111	110	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s )))) = (2s ·s ((𝐴 /su (2s↑s𝑦)) /su 2s)))
112	69, 103, 107	divscld 28283	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s𝑦)) ∈ No )
113	112, 104, 108	divscan2d 28284	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s ((𝐴 /su (2s↑s𝑦)) /su 2s)) = (𝐴 /su (2s↑s𝑦)))
114	111, 113	eqtrd 2787	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s )))) = (𝐴 /su (2s↑s𝑦)))
115	114	sneqd 4584	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → {(2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s ))))} = {(𝐴 /su (2s↑s𝑦))})
116	99	oveq2d 7397	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) = (𝐵 /su ((2s↑s𝑦) ·s 2s)))
117	81, 103, 104, 107, 108	divdivs1d 28292	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐵 /su (2s↑s𝑦)) /su 2s) = (𝐵 /su ((2s↑s𝑦) ·s 2s)))
118	116, 117	eqtr4d 2790	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) = ((𝐵 /su (2s↑s𝑦)) /su 2s))
119	118	oveq2d 7397	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s )))) = (2s ·s ((𝐵 /su (2s↑s𝑦)) /su 2s)))
120	81, 103, 107	divscld 28283	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s𝑦)) ∈ No )
121	120, 104, 108	divscan2d 28284	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s ((𝐵 /su (2s↑s𝑦)) /su 2s)) = (𝐵 /su (2s↑s𝑦)))
122	119, 121	eqtrd 2787	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s )))) = (𝐵 /su (2s↑s𝑦)))
123	122	sneqd 4584	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → {(2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s ))))} = {(𝐵 /su (2s↑s𝑦))})
124	115, 123	oveq12d 7399	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ({(2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s ))))} |s {(2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s ))))}) = ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}))
125	124	eqcomd 2758	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ({(2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s ))))} |s {(2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s ))))}))
126	69, 81, 73, 78	divsdird 28294	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) = ((𝐴 /su (2s↑s(𝑦 +s 1s ))) +s (𝐵 /su (2s↑s(𝑦 +s 1s )))))
127	125, 126	eqeq12d 2768	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) ↔ ({(2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s ))))} |s {(2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s ))))}) = ((𝐴 /su (2s↑s(𝑦 +s 1s ))) +s (𝐵 /su (2s↑s(𝑦 +s 1s ))))))
128	127	biimp3a 1480	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → ({(2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s ))))} |s {(2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s ))))}) = ((𝐴 /su (2s↑s(𝑦 +s 1s ))) +s (𝐵 /su (2s↑s(𝑦 +s 1s )))))
129		eqid 2752	. . . . . . . 8 ⊢ ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))})
130	80, 83, 96, 128, 129	halfcut 28517	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = (((𝐴 /su (2s↑s(𝑦 +s 1s ))) +s (𝐵 /su (2s↑s(𝑦 +s 1s )))) /su 2s))
131	126	oveq1d 7396	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) /su 2s) = (((𝐴 /su (2s↑s(𝑦 +s 1s ))) +s (𝐵 /su (2s↑s(𝑦 +s 1s )))) /su 2s))
132	131	3adant3 1141	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → (((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) /su 2s) = (((𝐴 /su (2s↑s(𝑦 +s 1s ))) +s (𝐵 /su (2s↑s(𝑦 +s 1s )))) /su 2s))
133	130, 132	eqtr4d 2790	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = (((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) /su 2s))
134		expsp1 28488	. . . . . . . . . . 11 ⊢ ((2s ∈ No ∧ (𝑦 +s 1s ) ∈ ℕ0s) → (2s↑s((𝑦 +s 1s ) +s 1s )) = ((2s↑s(𝑦 +s 1s )) ·s 2s))
135	3, 70, 134	sylancr 595	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → (2s↑s((𝑦 +s 1s ) +s 1s )) = ((2s↑s(𝑦 +s 1s )) ·s 2s))
136	135	adantr 483	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s((𝑦 +s 1s ) +s 1s )) = ((2s↑s(𝑦 +s 1s )) ·s 2s))
137	136	oveq2d 7397	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))) = ((𝐴 +s 𝐵) /su ((2s↑s(𝑦 +s 1s )) ·s 2s)))
138	69, 81	addscld 28039	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 +s 𝐵) ∈ No )
139	138, 73, 104, 78, 108	divdivs1d 28292	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) /su 2s) = ((𝐴 +s 𝐵) /su ((2s↑s(𝑦 +s 1s )) ·s 2s)))
140	137, 139	eqtr4d 2790	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))) = (((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) /su 2s))
141	140	3adant3 1141	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))) = (((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) /su 2s))
142	133, 141	eqtr4d 2790	. . . . 5 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑 ∧ ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))))
143	142	3exp 1128	. . . 4 ⊢ (𝑦 ∈ ℕ0s → (𝜑 → (({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s ))) → ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))))))
144	143	a2d 29	. . 3 ⊢ (𝑦 ∈ ℕ0s → ((𝜑 → ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑦 +s 1s )))) → (𝜑 → ({(𝐴 /su (2s↑s(𝑦 +s 1s )))} |s {(𝐵 /su (2s↑s(𝑦 +s 1s )))}) = ((𝐴 +s 𝐵) /su (2s↑s((𝑦 +s 1s ) +s 1s ))))))
145	23, 34, 45, 56, 68, 144	n0sind 28392	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s )))))
146	1, 145	mpcom 38	1 ⊢ (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s ))))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1550   ∈ wcel 2132   ≠ wne 2947  {csn 4572   class class class wbr 5090  (class class class)co 7381   No csur 27670   <s clts 27671   |s ccuts 27818   0_s c0s 27864   1_s c1s 27865   +_s cadds 28018   ·_s cmuls 28165   /_su cdivs 28246  ℕ_0scn0s 28371  ℕ_scnns 28372  2_sc2s 28469  ↑_scexps 28471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-dc 10389

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-pss 3915  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-tp 4577  df-op 4579  df-ot 4581  df-uni 4856  df-int 4896  df-iun 4941  df-br 5091  df-opab 5153  df-mpt 5172  df-tr 5198  df-id 5531  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5589  df-se 5590  df-we 5591  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-pred 6273  df-ord 6334  df-on 6335  df-lim 6336  df-suc 6337  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-om 7832  df-1st 7955  df-2nd 7956  df-frecs 8246  df-wrecs 8277  df-recs 8326  df-rdg 8365  df-1o 8421  df-2o 8422  df-oadd 8425  df-nadd 8620  df-no 27673  df-lts 27674  df-bday 27675  df-les 27775  df-slts 27817  df-cuts 27819  df-0s 27866  df-1s 27867  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27997  df-norec2 28008  df-adds 28019  df-negs 28080  df-subs 28081  df-muls 28166  df-divs 28247  df-seqs 28343  df-n0s 28373  df-nns 28374  df-zs 28438  df-2s 28470  df-exps 28472

This theorem is referenced by:  pw2cutp1  28520