Theorem cutpw2 28432

Description: A cut expression for inverses of powers of two. (Contributed by Scott Fenton, 7-Aug-2025.)

Assertion

Ref	Expression
cutpw2	⊢ (𝑁 ∈ ℕ0s → ( 1s /su (2s↑s(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑁))}))

Proof of Theorem cutpw2

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

Step	Hyp	Ref	Expression
1		oveq1 7438	. . . . . . 7 ⊢ (𝑚 = 0s → (𝑚 +s 1s ) = ( 0s +s 1s ))
2		1sno 27887	. . . . . . . 8 ⊢ 1s ∈ No
3		addslid 28016	. . . . . . . 8 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ ( 0s +s 1s ) = 1s
5	1, 4	eqtrdi 2791	. . . . . 6 ⊢ (𝑚 = 0s → (𝑚 +s 1s ) = 1s )
6	5	oveq2d 7447	. . . . 5 ⊢ (𝑚 = 0s → (2s↑s(𝑚 +s 1s )) = (2s↑s 1s ))
7		2sno 28418	. . . . . 6 ⊢ 2s ∈ No
8		exps1 28426	. . . . . 6 ⊢ (2s ∈ No → (2s↑s 1s ) = 2s)
9	7, 8	ax-mp 5	. . . . 5 ⊢ (2s↑s 1s ) = 2s
10	6, 9	eqtrdi 2791	. . . 4 ⊢ (𝑚 = 0s → (2s↑s(𝑚 +s 1s )) = 2s)
11	10	oveq2d 7447	. . 3 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s(𝑚 +s 1s ))) = ( 1s /su 2s))
12		oveq2 7439	. . . . . . . 8 ⊢ (𝑚 = 0s → (2s↑s𝑚) = (2s↑s 0s ))
13		exps0 28425	. . . . . . . . 9 ⊢ (2s ∈ No → (2s↑s 0s ) = 1s )
14	7, 13	ax-mp 5	. . . . . . . 8 ⊢ (2s↑s 0s ) = 1s
15	12, 14	eqtrdi 2791	. . . . . . 7 ⊢ (𝑚 = 0s → (2s↑s𝑚) = 1s )
16	15	oveq2d 7447	. . . . . 6 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = ( 1s /su 1s ))
17		divs1 28244	. . . . . . 7 ⊢ ( 1s ∈ No → ( 1s /su 1s ) = 1s )
18	2, 17	ax-mp 5	. . . . . 6 ⊢ ( 1s /su 1s ) = 1s
19	16, 18	eqtrdi 2791	. . . . 5 ⊢ (𝑚 = 0s → ( 1s /su (2s↑s𝑚)) = 1s )
20	19	sneqd 4643	. . . 4 ⊢ (𝑚 = 0s → {( 1s /su (2s↑s𝑚))} = { 1s })
21	20	oveq2d 7447	. . 3 ⊢ (𝑚 = 0s → ({ 0s } |s {( 1s /su (2s↑s𝑚))}) = ({ 0s } |s { 1s }))
22	11, 21	eqeq12d 2751	. 2 ⊢ (𝑚 = 0s → (( 1s /su (2s↑s(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑚))}) ↔ ( 1s /su 2s) = ({ 0s } |s { 1s })))
23		oveq1 7438	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 +s 1s ) = (𝑛 +s 1s ))
24	23	oveq2d 7447	. . . 4 ⊢ (𝑚 = 𝑛 → (2s↑s(𝑚 +s 1s )) = (2s↑s(𝑛 +s 1s )))
25	24	oveq2d 7447	. . 3 ⊢ (𝑚 = 𝑛 → ( 1s /su (2s↑s(𝑚 +s 1s ))) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
26		oveq2 7439	. . . . . 6 ⊢ (𝑚 = 𝑛 → (2s↑s𝑚) = (2s↑s𝑛))
27	26	oveq2d 7447	. . . . 5 ⊢ (𝑚 = 𝑛 → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s𝑛)))
28	27	sneqd 4643	. . . 4 ⊢ (𝑚 = 𝑛 → {( 1s /su (2s↑s𝑚))} = {( 1s /su (2s↑s𝑛))})
29	28	oveq2d 7447	. . 3 ⊢ (𝑚 = 𝑛 → ({ 0s } |s {( 1s /su (2s↑s𝑚))}) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}))
30	25, 29	eqeq12d 2751	. 2 ⊢ (𝑚 = 𝑛 → (( 1s /su (2s↑s(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑚))}) ↔ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})))
31		oveq1 7438	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑚 +s 1s ) = ((𝑛 +s 1s ) +s 1s ))
32	31	oveq2d 7447	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s(𝑚 +s 1s )) = (2s↑s((𝑛 +s 1s ) +s 1s )))
33	32	oveq2d 7447	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2s↑s(𝑚 +s 1s ))) = ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))))
34		oveq2 7439	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (2s↑s𝑚) = (2s↑s(𝑛 +s 1s )))
35	34	oveq2d 7447	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
36	35	sneqd 4643	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → {( 1s /su (2s↑s𝑚))} = {( 1s /su (2s↑s(𝑛 +s 1s )))})
37	36	oveq2d 7447	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → ({ 0s } |s {( 1s /su (2s↑s𝑚))}) = ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))}))
38	33, 37	eqeq12d 2751	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (( 1s /su (2s↑s(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑚))}) ↔ ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))})))
39		oveq1 7438	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 +s 1s ) = (𝑁 +s 1s ))
40	39	oveq2d 7447	. . . 4 ⊢ (𝑚 = 𝑁 → (2s↑s(𝑚 +s 1s )) = (2s↑s(𝑁 +s 1s )))
41	40	oveq2d 7447	. . 3 ⊢ (𝑚 = 𝑁 → ( 1s /su (2s↑s(𝑚 +s 1s ))) = ( 1s /su (2s↑s(𝑁 +s 1s ))))
42		oveq2 7439	. . . . . 6 ⊢ (𝑚 = 𝑁 → (2s↑s𝑚) = (2s↑s𝑁))
43	42	oveq2d 7447	. . . . 5 ⊢ (𝑚 = 𝑁 → ( 1s /su (2s↑s𝑚)) = ( 1s /su (2s↑s𝑁)))
44	43	sneqd 4643	. . . 4 ⊢ (𝑚 = 𝑁 → {( 1s /su (2s↑s𝑚))} = {( 1s /su (2s↑s𝑁))})
45	44	oveq2d 7447	. . 3 ⊢ (𝑚 = 𝑁 → ({ 0s } |s {( 1s /su (2s↑s𝑚))}) = ({ 0s } |s {( 1s /su (2s↑s𝑁))}))
46	41, 45	eqeq12d 2751	. 2 ⊢ (𝑚 = 𝑁 → (( 1s /su (2s↑s(𝑚 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑚))}) ↔ ( 1s /su (2s↑s(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑁))})))
47		nohalf 28422	. 2 ⊢ ( 1s /su 2s) = ({ 0s } |s { 1s })
48		peano2n0s 28350	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (𝑛 +s 1s ) ∈ ℕ0s)
49		expsp1 28427	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) ·s 2s))
50	7, 49	mpan 690	. . . . . . . . 9 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) ·s 2s))
51	48, 50	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (2s↑s((𝑛 +s 1s ) +s 1s )) = ((2s↑s(𝑛 +s 1s )) ·s 2s))
52	51	oveq2d 7447	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = ( 1s /su ((2s↑s(𝑛 +s 1s )) ·s 2s)))
53	2	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 1s ∈ No )
54		expscl 28428	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) ∈ No )
55	7, 54	mpan 690	. . . . . . . . 9 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) ∈ No )
56	48, 55	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) ∈ No )
57	7	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 2s ∈ No )
58		2ne0s 28419	. . . . . . . . . 10 ⊢ 2s ≠ 0s
59		expsne0 28429	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ (𝑛 +s 1s ) ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) ≠ 0s )
60	7, 58, 59	mp3an12 1450	. . . . . . . . 9 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) ≠ 0s )
61	48, 60	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) ≠ 0s )
62	58	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 2s ≠ 0s )
63	53, 56, 57, 61, 62	divdivs1d 28272	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → (( 1s /su (2s↑s(𝑛 +s 1s ))) /su 2s) = ( 1s /su ((2s↑s(𝑛 +s 1s )) ·s 2s)))
64	52, 63	eqtr4d 2778	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = (( 1s /su (2s↑s(𝑛 +s 1s ))) /su 2s))
65	53, 56, 61	divscld 28263	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s(𝑛 +s 1s ))) ∈ No )
66		addslid 28016	. . . . . . . 8 ⊢ (( 1s /su (2s↑s(𝑛 +s 1s ))) ∈ No → ( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
67	65, 66	syl 17	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
68	67	oveq1d 7446	. . . . . 6 ⊢ (𝑛 ∈ ℕ0s → (( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) /su 2s) = (( 1s /su (2s↑s(𝑛 +s 1s ))) /su 2s))
69	64, 68	eqtr4d 2778	. . . . 5 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = (( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) /su 2s))
70	69	adantr 480	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = (( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) /su 2s))
71		0sno 27886	. . . . . 6 ⊢ 0s ∈ No
72	71	a1i 11	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → 0s ∈ No )
73	2	a1i 11	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → 1s ∈ No )
74	56	adantr 480	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → (2s↑s(𝑛 +s 1s )) ∈ No )
75	61	adantr 480	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → (2s↑s(𝑛 +s 1s )) ≠ 0s )
76	73, 74, 75	divscld 28263	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ( 1s /su (2s↑s(𝑛 +s 1s ))) ∈ No )
77		muls02 28182	. . . . . . . 8 ⊢ ((2s↑s(𝑛 +s 1s )) ∈ No → ( 0s ·s (2s↑s(𝑛 +s 1s ))) = 0s )
78	74, 77	syl 17	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ( 0s ·s (2s↑s(𝑛 +s 1s ))) = 0s )
79		0slt1s 27889	. . . . . . 7 ⊢ 0s <s 1s
80	78, 79	eqbrtrdi 5187	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ( 0s ·s (2s↑s(𝑛 +s 1s ))) <s 1s )
81		2nns 28417	. . . . . . . . . . 11 ⊢ 2s ∈ ℕs
82		nnsgt0 28357	. . . . . . . . . . 11 ⊢ (2s ∈ ℕs → 0s <s 2s)
83	81, 82	ax-mp 5	. . . . . . . . . 10 ⊢ 0s <s 2s
84		expsgt0 28430	. . . . . . . . . 10 ⊢ ((2s ∈ No ∧ (𝑛 +s 1s ) ∈ ℕ0s ∧ 0s <s 2s) → 0s <s (2s↑s(𝑛 +s 1s )))
85	7, 83, 84	mp3an13 1451	. . . . . . . . 9 ⊢ ((𝑛 +s 1s ) ∈ ℕ0s → 0s <s (2s↑s(𝑛 +s 1s )))
86	48, 85	syl 17	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → 0s <s (2s↑s(𝑛 +s 1s )))
87	86	adantr 480	. . . . . . 7 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → 0s <s (2s↑s(𝑛 +s 1s )))
88	72, 73, 74, 87	sltmuldivd 28268	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → (( 0s ·s (2s↑s(𝑛 +s 1s ))) <s 1s ↔ 0s <s ( 1s /su (2s↑s(𝑛 +s 1s )))))
89	80, 88	mpbid 232	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → 0s <s ( 1s /su (2s↑s(𝑛 +s 1s ))))
90		muls01 28153	. . . . . . . . . . 11 ⊢ (2s ∈ No → (2s ·s 0s ) = 0s )
91	7, 90	ax-mp 5	. . . . . . . . . 10 ⊢ (2s ·s 0s ) = 0s
92	91	sneqi 4642	. . . . . . . . 9 ⊢ {(2s ·s 0s )} = { 0s }
93	92	a1i 11	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → {(2s ·s 0s )} = { 0s })
94		expsp1 28427	. . . . . . . . . . . . . 14 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
95	7, 94	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0s → (2s↑s(𝑛 +s 1s )) = ((2s↑s𝑛) ·s 2s))
96	95	oveq2d 7447	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s(𝑛 +s 1s ))) = ( 1s /su ((2s↑s𝑛) ·s 2s)))
97		expscl 28428	. . . . . . . . . . . . . 14 ⊢ ((2s ∈ No ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ∈ No )
98	7, 97	mpan 690	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ∈ No )
99		expsne0 28429	. . . . . . . . . . . . . 14 ⊢ ((2s ∈ No ∧ 2s ≠ 0s ∧ 𝑛 ∈ ℕ0s) → (2s↑s𝑛) ≠ 0s )
100	7, 58, 99	mp3an12 1450	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ0s → (2s↑s𝑛) ≠ 0s )
101	53, 98, 57, 100, 62	divdivs1d 28272	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ0s → (( 1s /su (2s↑s𝑛)) /su 2s) = ( 1s /su ((2s↑s𝑛) ·s 2s)))
102	96, 101	eqtr4d 2778	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s(𝑛 +s 1s ))) = (( 1s /su (2s↑s𝑛)) /su 2s))
103	102	oveq2d 7447	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → (2s ·s ( 1s /su (2s↑s(𝑛 +s 1s )))) = (2s ·s (( 1s /su (2s↑s𝑛)) /su 2s)))
104	53, 98, 100	divscld 28263	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0s → ( 1s /su (2s↑s𝑛)) ∈ No )
105	104, 57, 62	divscan2d 28264	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0s → (2s ·s (( 1s /su (2s↑s𝑛)) /su 2s)) = ( 1s /su (2s↑s𝑛)))
106	103, 105	eqtrd 2775	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0s → (2s ·s ( 1s /su (2s↑s(𝑛 +s 1s )))) = ( 1s /su (2s↑s𝑛)))
107	106	sneqd 4643	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0s → {(2s ·s ( 1s /su (2s↑s(𝑛 +s 1s ))))} = {( 1s /su (2s↑s𝑛))})
108	93, 107	oveq12d 7449	. . . . . . 7 ⊢ (𝑛 ∈ ℕ0s → ({(2s ·s 0s )} |s {(2s ·s ( 1s /su (2s↑s(𝑛 +s 1s ))))}) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}))
109		eqcom 2742	. . . . . . . 8 ⊢ (( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}) ↔ ({ 0s } |s {( 1s /su (2s↑s𝑛))}) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
110	109	biimpi 216	. . . . . . 7 ⊢ (( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}) → ({ 0s } |s {( 1s /su (2s↑s𝑛))}) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
111	108, 110	sylan9eq 2795	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ({(2s ·s 0s )} |s {(2s ·s ( 1s /su (2s↑s(𝑛 +s 1s ))))}) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
112	76, 66	syl 17	. . . . . 6 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) = ( 1s /su (2s↑s(𝑛 +s 1s ))))
113	111, 112	eqtr4d 2778	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ({(2s ·s 0s )} |s {(2s ·s ( 1s /su (2s↑s(𝑛 +s 1s ))))}) = ( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))))
114		eqid 2735	. . . . 5 ⊢ ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))}) = ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))})
115	72, 76, 89, 113, 114	halfcut 28431	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))}) = (( 0s +s ( 1s /su (2s↑s(𝑛 +s 1s )))) /su 2s))
116	70, 115	eqtr4d 2778	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))})) → ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))}))
117	116	ex 412	. 2 ⊢ (𝑛 ∈ ℕ0s → (( 1s /su (2s↑s(𝑛 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑛))}) → ( 1s /su (2s↑s((𝑛 +s 1s ) +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s(𝑛 +s 1s )))})))
118	22, 30, 38, 46, 47, 117	n0sind 28352	1 ⊢ (𝑁 ∈ ℕ0s → ( 1s /su (2s↑s(𝑁 +s 1s ))) = ({ 0s } |s {( 1s /su (2s↑s𝑁))}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  {csn 4631   class class class wbr 5148  (class class class)co 7431   No csur 27699   <s cslt 27700   |s cscut 27842   0_s c0s 27882   1_s c1s 27883   +_s cadds 28007   ·_s cmuls 28147   /_su cdivs 28228  ℕ_0scnn0s 28333  ℕ_scnns 28334  2_sc2s 28409  ↑_scexps 28411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-dc 10484

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-muls 28148  df-divs 28229  df-seqs 28305  df-n0s 28335  df-nns 28336  df-zs 28380  df-2s 28410  df-exps 28412

This theorem is referenced by:  pw2bday  28433