Theorem pw2cut 28335

Description: Extend halfcut 28333 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 7395	. . . . . . . . 9 ⊢ (𝑥 = 0s → (2s↑s𝑥) = (2s↑s 0s ))
3		2sno 28305	. . . . . . . . . 10 ⊢ 2s ∈ No
4		exps0 28313	. . . . . . . . . 10 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
5	3, 4	ax-mp 5	. . . . . . . . 9 ⊢ (2s↑s 0s ) = 1s
6	2, 5	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑥 = 0s → (2s↑s𝑥) = 1s )
7	6	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 0s → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su 1s ))
8	7	sneqd 4601	. . . . . 6 ⊢ (𝑥 = 0s → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su 1s )})
9	6	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 0s → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su 1s ))
10	9	sneqd 4601	. . . . . 6 ⊢ (𝑥 = 0s → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su 1s )})
11	8, 10	oveq12d 7405	. . . . 5 ⊢ (𝑥 = 0s → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}))
12		oveq1 7394	. . . . . . . . 9 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = ( 0s +s 1s ))
13		1sno 27739	. . . . . . . . . 10 ⊢ 1s ∈ No
14		addslid 27875	. . . . . . . . . 10 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
15	13, 14	ax-mp 5	. . . . . . . . 9 ⊢ ( 0s +s 1s ) = 1s
16	12, 15	eqtrdi 2780	. . . . . . . 8 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = 1s )
17	16	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 0s → (2s↑s(𝑥 +s 1s )) = (2s↑s 1s ))
18		exps1 28314	. . . . . . . 8 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
19	3, 18	ax-mp 5	. . . . . . 7 ⊢ (2s↑s 1s ) = 2s
20	17, 19	eqtrdi 2780	. . . . . 6 ⊢ (𝑥 = 0s → (2s↑s(𝑥 +s 1s )) = 2s)
21	20	oveq2d 7403	. . . . 5 ⊢ (𝑥 = 0s → ((𝐴 +s 𝐵) /su (2s↑s(𝑥 +s 1s ))) = ((𝐴 +s 𝐵) /su 2s))
22	11, 21	eqeq12d 2745	. . . 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 340	. . 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 7395	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (2s↑s𝑥) = (2s↑s𝑦))
25	24	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su (2s↑s𝑦)))
26	25	sneqd 4601	. . . . . 6 ⊢ (𝑥 = 𝑦 → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su (2s↑s𝑦))})
27	24	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su (2s↑s𝑦)))
28	27	sneqd 4601	. . . . . 6 ⊢ (𝑥 = 𝑦 → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su (2s↑s𝑦))})
29	26, 28	oveq12d 7405	. . . . 5 ⊢ (𝑥 = 𝑦 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su (2s↑s𝑦))} |s {(𝐵 /su (2s↑s𝑦))}))
30		oveq1 7394	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
31	30	oveq2d 7403	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2s↑s(𝑥 +s 1s )) = (2s↑s(𝑦 +s 1s )))
32	31	oveq2d 7403	. . . . 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 2745	. . . 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 340	. . 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 7395	. . . . . . . 8 ⊢ (𝑥 = (𝑦 +s 1s ) → (2s↑s𝑥) = (2s↑s(𝑦 +s 1s )))
36	35	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su (2s↑s(𝑦 +s 1s ))))
37	36	sneqd 4601	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su (2s↑s(𝑦 +s 1s )))})
38	35	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su (2s↑s(𝑦 +s 1s ))))
39	38	sneqd 4601	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su (2s↑s(𝑦 +s 1s )))})
40	37, 39	oveq12d 7405	. . . . 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 7394	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 +s 1s ) = ((𝑦 +s 1s ) +s 1s ))
42	41	oveq2d 7403	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (2s↑s(𝑥 +s 1s )) = (2s↑s((𝑦 +s 1s ) +s 1s )))
43	42	oveq2d 7403	. . . . 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 2745	. . . 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 340	. . 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 7395	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (2s↑s𝑥) = (2s↑s𝑁))
47	46	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐴 /su (2s↑s𝑥)) = (𝐴 /su (2s↑s𝑁)))
48	47	sneqd 4601	. . . . . 6 ⊢ (𝑥 = 𝑁 → {(𝐴 /su (2s↑s𝑥))} = {(𝐴 /su (2s↑s𝑁))})
49	46	oveq2d 7403	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝐵 /su (2s↑s𝑥)) = (𝐵 /su (2s↑s𝑁)))
50	49	sneqd 4601	. . . . . 6 ⊢ (𝑥 = 𝑁 → {(𝐵 /su (2s↑s𝑥))} = {(𝐵 /su (2s↑s𝑁))})
51	48, 50	oveq12d 7405	. . . . 5 ⊢ (𝑥 = 𝑁 → ({(𝐴 /su (2s↑s𝑥))} |s {(𝐵 /su (2s↑s𝑥))}) = ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}))
52		oveq1 7394	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 +s 1s ) = (𝑁 +s 1s ))
53	52	oveq2d 7403	. . . . . 6 ⊢ (𝑥 = 𝑁 → (2s↑s(𝑥 +s 1s )) = (2s↑s(𝑁 +s 1s )))
54	53	oveq2d 7403	. . . . 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 2745	. . . 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 340	. . 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		divs1 28107	. . . . . . 7 ⊢ (𝐴 ∈ No → (𝐴 /su 1s ) = 𝐴)
59	57, 58	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐴 /su 1s ) = 𝐴)
60	59	sneqd 4601	. . . . 5 ⊢ (𝜑 → {(𝐴 /su 1s )} = {𝐴})
61		pw2cut.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ No )
62		divs1 28107	. . . . . . 7 ⊢ (𝐵 ∈ No → (𝐵 /su 1s ) = 𝐵)
63	61, 62	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐵 /su 1s ) = 𝐵)
64	63	sneqd 4601	. . . . 5 ⊢ (𝜑 → {(𝐵 /su 1s )} = {𝐵})
65	60, 64	oveq12d 7405	. . . 4 ⊢ (𝜑 → ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}) = ({𝐴} |s {𝐵}))
66		pw2cut.4	. . . . 5 ⊢ (𝜑 → 𝐴 <s 𝐵)
67		pw2cut.5	. . . . 5 ⊢ (𝜑 → ({(2s ·s 𝐴)} |s {(2s ·s 𝐵)}) = (𝐴 +s 𝐵))
68		eqid 2729	. . . . 5 ⊢ ({𝐴} |s {𝐵}) = ({𝐴} |s {𝐵})
69	57, 61, 66, 67, 68	halfcut 28333	. . . 4 ⊢ (𝜑 → ({𝐴} |s {𝐵}) = ((𝐴 +s 𝐵) /su 2s))
70	65, 69	eqtrd 2764	. . 3 ⊢ (𝜑 → ({(𝐴 /su 1s )} |s {(𝐵 /su 1s )}) = ((𝐴 +s 𝐵) /su 2s))
71	57	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝐴 ∈ No )
72		peano2n0s 28223	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
73		expscl 28317	. . . . . . . . . . . 12 ⊢ ((2s ∈ No ∧ (𝑦 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑦 +s 1s )) ∈ No )
74	3, 72, 73	sylancr 587	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) ∈ No )
75	74	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s(𝑦 +s 1s )) ∈ No )
76		2ne0s 28306	. . . . . . . . . . . . 13 ⊢ 2s ≠ 0s
77		expsne0 28321	. . . . . . . . . . . . 13 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ (𝑦 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑦 +s 1s )) ≠ 0s )
78	3, 76, 77	mp3an12 1453	. . . . . . . . . . . 12 ⊢ ((𝑦 +s 1s ) ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) ≠ 0s )
79	72, 78	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) ≠ 0s )
80	79	adantr 480	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s(𝑦 +s 1s )) ≠ 0s )
81	71, 75, 80	divscld 28126	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) ∈ No )
82	81	3adant3 1132	. . . . . . . 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 )
83	61	adantl 481	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝐵 ∈ No )
84	83, 75, 80	divscld 28126	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) ∈ No )
85	84	3adant3 1132	. . . . . . . 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 )
86	71, 75, 80	divscan1d 28128	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 /su (2s↑s(𝑦 +s 1s ))) ·s (2s↑s(𝑦 +s 1s ))) = 𝐴)
87	66	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 𝐴 <s 𝐵)
88	86, 87	eqbrtrd 5129	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 /su (2s↑s(𝑦 +s 1s ))) ·s (2s↑s(𝑦 +s 1s ))) <s 𝐵)
89		2nns 28304	. . . . . . . . . . . . . . 15 ⊢ 2s ∈ ℕs
90		nnsgt0 28231	. . . . . . . . . . . . . . 15 ⊢ (2s ∈ ℕs → 0s <s 2s)
91	89, 90	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ 0s <s 2s
92		expsgt0 28322	. . . . . . . . . . . . . 14 ⊢ ((2s ∈ No ∧ (𝑦 +s 1s ) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑦 +s 1s )))
93	3, 91, 92	mp3an13 1454	. . . . . . . . . . . . 13 ⊢ ((𝑦 +s 1s ) ∈ ℕ0s → 0s <s (2s↑s(𝑦 +s 1s )))
94	72, 93	syl 17	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 0s <s (2s↑s(𝑦 +s 1s )))
95	94	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 0s <s (2s↑s(𝑦 +s 1s )))
96	81, 83, 75, 95	sltmuldivd 28131	. . . . . . . . . 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 )))))
97	88, 96	mpbid 232	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) <s (𝐵 /su (2s↑s(𝑦 +s 1s ))))
98	97	3adant3 1132	. . . . . . . 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 ))))
99		expsp1 28315	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑦 ∈ ℕ0s) → (2s↑s(𝑦 +s 1s )) = ((2s↑s𝑦) ·s 2s))
100	3, 99	mpan 690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0s → (2s↑s(𝑦 +s 1s )) = ((2s↑s𝑦) ·s 2s))
101	100	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s(𝑦 +s 1s )) = ((2s↑s𝑦) ·s 2s))
102	101	oveq2d 7403	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) = (𝐴 /su ((2s↑s𝑦) ·s 2s)))
103		expscl 28317	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 𝑦 ∈ ℕ0s) → (2s↑s𝑦) ∈ No )
104	3, 103	mpan 690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0s → (2s↑s𝑦) ∈ No )
105	104	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s𝑦) ∈ No )
106	3	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 2s ∈ No )
107		expsne0 28321	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ 𝑦 ∈ ℕ0s) → (2s↑s𝑦) ≠ 0s )
108	3, 76, 107	mp3an12 1453	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ℕ0s → (2s↑s𝑦) ≠ 0s )
109	108	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s𝑦) ≠ 0s )
110	76	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → 2s ≠ 0s )
111	71, 105, 106, 109, 110	divdivs1d 28135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐴 /su (2s↑s𝑦)) /su 2s) = (𝐴 /su ((2s↑s𝑦) ·s 2s)))
112	102, 111	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s(𝑦 +s 1s ))) = ((𝐴 /su (2s↑s𝑦)) /su 2s))
113	112	oveq2d 7403	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s )))) = (2s ·s ((𝐴 /su (2s↑s𝑦)) /su 2s)))
114	71, 105, 109	divscld 28126	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 /su (2s↑s𝑦)) ∈ No )
115	114, 106, 110	divscan2d 28127	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s ((𝐴 /su (2s↑s𝑦)) /su 2s)) = (𝐴 /su (2s↑s𝑦)))
116	113, 115	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s )))) = (𝐴 /su (2s↑s𝑦)))
117	116	sneqd 4601	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → {(2s ·s (𝐴 /su (2s↑s(𝑦 +s 1s ))))} = {(𝐴 /su (2s↑s𝑦))})
118	101	oveq2d 7403	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) = (𝐵 /su ((2s↑s𝑦) ·s 2s)))
119	83, 105, 106, 109, 110	divdivs1d 28135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → ((𝐵 /su (2s↑s𝑦)) /su 2s) = (𝐵 /su ((2s↑s𝑦) ·s 2s)))
120	118, 119	eqtr4d 2767	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s(𝑦 +s 1s ))) = ((𝐵 /su (2s↑s𝑦)) /su 2s))
121	120	oveq2d 7403	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s )))) = (2s ·s ((𝐵 /su (2s↑s𝑦)) /su 2s)))
122	83, 105, 109	divscld 28126	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐵 /su (2s↑s𝑦)) ∈ No )
123	122, 106, 110	divscan2d 28127	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s ((𝐵 /su (2s↑s𝑦)) /su 2s)) = (𝐵 /su (2s↑s𝑦)))
124	121, 123	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s )))) = (𝐵 /su (2s↑s𝑦)))
125	124	sneqd 4601	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → {(2s ·s (𝐵 /su (2s↑s(𝑦 +s 1s ))))} = {(𝐵 /su (2s↑s𝑦))})
126	117, 125	oveq12d 7405	. . . . . . . . . . 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))}))
127	126	eqcomd 2735	. . . . . . . . . 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 ))))}))
128	71, 83, 75, 80	divsdird 28137	. . . . . . . . . 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 )))))
129	127, 128	eqeq12d 2745	. . . . . . . . 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 ))))))
130	129	biimp3a 1471	. . . . . . . 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 )))))
131		eqid 2729	. . . . . . . 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 )))})
132	82, 85, 98, 130, 131	halfcut 28333	. . . . . . 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))
133	128	oveq1d 7402	. . . . . . . 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))
134	133	3adant3 1132	. . . . . . 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))
135	132, 134	eqtr4d 2767	. . . . . 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))
136		expsp1 28315	. . . . . . . . . . 11 ⊢ ((2s ∈ No ∧ (𝑦 +s 1s ) ∈ ℕ0s) → (2s↑s((𝑦 +s 1s ) +s 1s )) = ((2s↑s(𝑦 +s 1s )) ·s 2s))
137	3, 72, 136	sylancr 587	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → (2s↑s((𝑦 +s 1s ) +s 1s )) = ((2s↑s(𝑦 +s 1s )) ·s 2s))
138	137	adantr 480	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (2s↑s((𝑦 +s 1s ) +s 1s )) = ((2s↑s(𝑦 +s 1s )) ·s 2s))
139	138	oveq2d 7403	. . . . . . . 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)))
140	71, 83	addscld 27887	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝜑) → (𝐴 +s 𝐵) ∈ No )
141	140, 75, 106, 80, 110	divdivs1d 28135	. . . . . . . 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)))
142	139, 141	eqtr4d 2767	. . . . . . 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))
143	142	3adant3 1132	. . . . . 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))
144	135, 143	eqtr4d 2767	. . . . 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 ))))
145	144	3exp 1119	. . . 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 ))))))
146	145	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 ))))))
147	23, 34, 45, 56, 70, 146	n0sind 28225	. 2 ⊢ (𝑁 ∈ ℕ0s → (𝜑 → ({(𝐴 /su (2s↑s𝑁))} |s {(𝐵 /su (2s↑s𝑁))}) = ((𝐴 +s 𝐵) /su (2s↑s(𝑁 +s 1s )))))
148	1, 147	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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {csn 4589   class class class wbr 5107  (class class class)co 7387   No csur 27551   <s cslt 27552   |s cscut 27694   0_s c0s 27734   1_s c1s 27735   +_s cadds 27866   ·_s cmuls 28009   /_su cdivs 28090  ℕ_0scnn0s 28206  ℕ_scnns 28207  2_sc2s 28296  ↑_scexps 28298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-dc 10399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-ot 4598  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-oadd 8438  df-nadd 8630  df-no 27554  df-slt 27555  df-bday 27556  df-sle 27657  df-sslt 27693  df-scut 27695  df-0s 27736  df-1s 27737  df-made 27755  df-old 27756  df-left 27758  df-right 27759  df-norec 27845  df-norec2 27856  df-adds 27867  df-negs 27927  df-subs 27928  df-muls 28010  df-divs 28091  df-seqs 28178  df-n0s 28208  df-nns 28209  df-zs 28267  df-2s 28297  df-exps 28299

This theorem is referenced by:  pw2cutp1  28336  zs12bday  28343