bitsfzo - Intuitionistic Logic Explorer

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Theorem bitsfzo 12518

Description: The bits of a number are all at positions less than 𝑀 iff the number is nonnegative and less than 2↑𝑀. (Contributed by Mario Carneiro, 5-Sep-2016.) (Proof shortened by AV, 1-Oct-2020.)

**Assertion**
Ref	Expression
bitsfzo	⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) → (𝑁 ∈ (0..^(2↑𝑀)) ↔ (bits‘𝑁) ⊆ (0..^𝑀)))

Proof of Theorem bitsfzo Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bitsval 12506	. . . 4 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
2		simp32 1060	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℕ₀)
3		nn0uz 9791	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2324	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (ℤ_≥‘0))
5		simp1r 1048	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℕ₀)
6	5	nn0zd 9600	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℤ)
7		2re 9213	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
8	7	a1i 9	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 2 ∈ ℝ)
9	8, 2	reexpcld 10953	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ)
10		simp1l 1047	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℤ)
11	10	zred 9602	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℝ)
12	8, 5	reexpcld 10953	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑀) ∈ ℝ)
13	9	recnd 8208	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℂ)
14	13	mullidd 8197	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) = (2↑𝑚))
15		1z 9505	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
16		zq 9860	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → 1 ∈ ℚ)
17	15, 16	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℚ
18	17	a1i 9	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ∈ ℚ)
19		2nn 9305	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℕ
20	19	a1i 9	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 2 ∈ ℕ)
21	20, 2	nnexpcld 10958	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℕ)
22		znq 9858	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ (2↑𝑚) ∈ ℕ) → (𝑁 / (2↑𝑚)) ∈ ℚ)
23	10, 21, 22	syl2anc 411	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (𝑁 / (2↑𝑚)) ∈ ℚ)
24		qdcle 10507	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℚ ∧ (𝑁 / (2↑𝑚)) ∈ ℚ) → DECID 1 ≤ (𝑁 / (2↑𝑚)))
25	18, 23, 24	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → DECID 1 ≤ (𝑁 / (2↑𝑚)))
26		simp33 1061	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
27		qltnle 10504	. . . . . . . . . . . . . 14 ⊢ (((𝑁 / (2↑𝑚)) ∈ ℚ ∧ 1 ∈ ℚ) → ((𝑁 / (2↑𝑚)) < 1 ↔ ¬ 1 ≤ (𝑁 / (2↑𝑚))))
28	23, 18, 27	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((𝑁 / (2↑𝑚)) < 1 ↔ ¬ 1 ≤ (𝑁 / (2↑𝑚))))
29		0p1e1 9257	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
30	29	breq2i 4096	. . . . . . . . . . . . . 14 ⊢ ((𝑁 / (2↑𝑚)) < (0 + 1) ↔ (𝑁 / (2↑𝑚)) < 1)
31		2rp 9893	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ⁺
32	31	a1i 9	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 2 ∈ ℝ⁺)
33	2	nn0zd 9600	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℤ)
34	32, 33	rpexpcld 10960	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ⁺)
35		elfzole1 10391	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 0 ≤ 𝑁)
36	35	3ad2ant2 1045	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ 𝑁)
37	11, 34, 36	divge0d 9972	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ (𝑁 / (2↑𝑚)))
38		0z 9490	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℤ
39		flqbi 10551	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 / (2↑𝑚)) ∈ ℚ ∧ 0 ∈ ℤ) → ((⌊‘(𝑁 / (2↑𝑚))) = 0 ↔ (0 ≤ (𝑁 / (2↑𝑚)) ∧ (𝑁 / (2↑𝑚)) < (0 + 1))))
40	23, 38, 39	sylancl 413	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((⌊‘(𝑁 / (2↑𝑚))) = 0 ↔ (0 ≤ (𝑁 / (2↑𝑚)) ∧ (𝑁 / (2↑𝑚)) < (0 + 1))))
41		z0even 12474	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∥ 0
42		id 19	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → (⌊‘(𝑁 / (2↑𝑚))) = 0)
43	41, 42	breqtrrid 4126	. . . . . . . . . . . . . . . 16 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
44	40, 43	biimtrrdi 164	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((0 ≤ (𝑁 / (2↑𝑚)) ∧ (𝑁 / (2↑𝑚)) < (0 + 1)) → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
45	37, 44	mpand 429	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((𝑁 / (2↑𝑚)) < (0 + 1) → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
46	30, 45	biimtrrid 153	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((𝑁 / (2↑𝑚)) < 1 → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
47	28, 46	sylbird 170	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (¬ 1 ≤ (𝑁 / (2↑𝑚)) → 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
48	26, 47	mtod 669	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ ¬ 1 ≤ (𝑁 / (2↑𝑚)))
49		notnotrdc 850	. . . . . . . . . . 11 ⊢ (DECID 1 ≤ (𝑁 / (2↑𝑚)) → (¬ ¬ 1 ≤ (𝑁 / (2↑𝑚)) → 1 ≤ (𝑁 / (2↑𝑚))))
50	25, 48, 49	sylc 62	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ≤ (𝑁 / (2↑𝑚)))
51		1red 8194	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ∈ ℝ)
52	51, 11, 34	lemuldivd 9981	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ((1 · (2↑𝑚)) ≤ 𝑁 ↔ 1 ≤ (𝑁 / (2↑𝑚))))
53	50, 52	mpbird 167	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) ≤ 𝑁)
54	14, 53	eqbrtrrd 4112	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ≤ 𝑁)
55		elfzolt2 10392	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 𝑁 < (2↑𝑀))
56	55	3ad2ant2 1045	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 < (2↑𝑀))
57	9, 11, 12, 54, 56	lelttrd 8304	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) < (2↑𝑀))
58		1lt2 9313	. . . . . . . . 9 ⊢ 1 < 2
59	58	a1i 9	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 < 2)
60		nn0ltexp2 10972	. . . . . . . 8 ⊢ (((2 ∈ ℝ ∧ 𝑚 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) ∧ 1 < 2) → (𝑚 < 𝑀 ↔ (2↑𝑚) < (2↑𝑀)))
61	8, 2, 5, 59, 60	syl31anc 1276	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (𝑚 < 𝑀 ↔ (2↑𝑚) < (2↑𝑀)))
62	57, 61	mpbird 167	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 < 𝑀)
63		elfzo2 10385	. . . . . 6 ⊢ (𝑚 ∈ (0..^𝑀) ↔ (𝑚 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ 𝑚 < 𝑀))
64	4, 6, 62, 63	syl3anbrc 1207	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (0..^𝑀))
65	64	3expia 1231	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → ((𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))) → 𝑚 ∈ (0..^𝑀)))
66	1, 65	biimtrid 152	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (𝑚 ∈ (bits‘𝑁) → 𝑚 ∈ (0..^𝑀)))
67	66	ssrdv 3233	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀))) → (bits‘𝑁) ⊆ (0..^𝑀))
68		simpr 110	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ)
69	68	nnred 9156	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℝ)
70		simpllr 536	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℕ₀)
71	70	nn0red 9456	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
72		maxle2 11774	. . . . . . 7 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → 𝑀 ≤ sup({-𝑁, 𝑀}, ℝ, < ))
73	69, 71, 72	syl2anc 411	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ≤ sup({-𝑁, 𝑀}, ℝ, < ))
74		simplr 529	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (bits‘𝑁) ⊆ (0..^𝑀))
75		n2dvdsm1 12476	. . . . . . . . . . 11 ⊢ ¬ 2 ∥ -1
76		simplll 535	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
77	76	zred 9602	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℝ)
78	19	a1i 9	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 2 ∈ ℕ)
79	68	nnnn0d 9455	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ₀)
80		nn0maxcl 11787	. . . . . . . . . . . . . . . . 17 ⊢ ((-𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℕ₀)
81	79, 70, 80	syl2anc 411	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℕ₀)
82	78, 81	nnexpcld 10958	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℕ)
83	77, 82	nndivred 9193	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∈ ℝ)
84		1red 8194	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 1 ∈ ℝ)
85	76	zcnd 9603	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
86	82	nncnd 9157	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℂ)
87	82	nnap0d 9189	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) # 0)
88	85, 86, 87	divnegapd 8983	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) = (-𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
89	81	nn0red 9456	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℝ)
90	82	nnred 9156	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℝ)
91		maxle1 11773	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-𝑁 ∈ ℝ ∧ 𝑀 ∈ ℝ) → -𝑁 ≤ sup({-𝑁, 𝑀}, ℝ, < ))
92	69, 71, 91	syl2anc 411	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ sup({-𝑁, 𝑀}, ℝ, < ))
93		2z 9507	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℤ
94		uzid 9770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
95	93, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ (ℤ_≥‘2)
96		bernneq3 10925	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℕ₀) → sup({-𝑁, 𝑀}, ℝ, < ) < (2↑sup({-𝑁, 𝑀}, ℝ, < )))
97	95, 81, 96	sylancr 414	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) < (2↑sup({-𝑁, 𝑀}, ℝ, < )))
98	89, 90, 97	ltled 8298	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ≤ (2↑sup({-𝑁, 𝑀}, ℝ, < )))
99	69, 89, 90, 92, 98	letrd 8303	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ (2↑sup({-𝑁, 𝑀}, ℝ, < )))
100	86	mulridd 8196	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((2↑sup({-𝑁, 𝑀}, ℝ, < )) · 1) = (2↑sup({-𝑁, 𝑀}, ℝ, < )))
101	99, 100	breqtrrd 4116	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ ((2↑sup({-𝑁, 𝑀}, ℝ, < )) · 1))
102	82	nnrpd 9929	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℝ⁺)
103	69, 84, 102	ledivmuld 9985	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((-𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ≤ 1 ↔ -𝑁 ≤ ((2↑sup({-𝑁, 𝑀}, ℝ, < )) · 1)))
104	101, 103	mpbird 167	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (-𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ≤ 1)
105	88, 104	eqbrtrd 4110	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ≤ 1)
106	83, 84, 105	lenegcon1d 8707	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -1 ≤ (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
107	68	nngt0d 9187	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -𝑁)
108	82	nngt0d 9187	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (2↑sup({-𝑁, 𝑀}, ℝ, < )))
109	69, 90, 107, 108	divgt0d 9115	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (-𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
110	109, 88	breqtrrd 4116	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
111	83	lt0neg1d 8695	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) < 0 ↔ 0 < -(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))))
112	110, 111	mpbird 167	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) < 0)
113		ax-1cn 8125	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
114		neg1cn 9248	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
115		1pneg1e0 9254	. . . . . . . . . . . . . . 15 ⊢ (1 + -1) = 0
116	113, 114, 115	addcomli 8324	. . . . . . . . . . . . . 14 ⊢ (-1 + 1) = 0
117	112, 116	breqtrrdi 4130	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) < (-1 + 1))
118		znq 9858	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℤ ∧ (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∈ ℚ)
119	76, 82, 118	syl2anc 411	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∈ ℚ)
120		neg1z 9511	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℤ
121		flqbi 10551	. . . . . . . . . . . . . 14 ⊢ (((𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∈ ℚ ∧ -1 ∈ ℤ) → ((⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))) = -1 ↔ (-1 ≤ (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∧ (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) < (-1 + 1))))
122	119, 120, 121	sylancl 413	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ((⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))) = -1 ↔ (-1 ≤ (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∧ (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) < (-1 + 1))))
123	106, 117, 122	mpbir2and 952	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))) = -1)
124	123	breq2d 4100	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2 ∥ (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))) ↔ 2 ∥ -1))
125	75, 124	mtbiri 681	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))))
126		bitsval2 12507	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℤ ∧ sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℕ₀) → (sup({-𝑁, 𝑀}, ℝ, < ) ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))))
127	76, 81, 126	syl2anc 411	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (sup({-𝑁, 𝑀}, ℝ, < ) ∈ (bits‘𝑁) ↔ ¬ 2 ∥ (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))))
128	125, 127	mpbird 167	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ (bits‘𝑁))
129	74, 128	sseldd 3228	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ (0..^𝑀))
130		elfzolt2 10392	. . . . . . . 8 ⊢ (sup({-𝑁, 𝑀}, ℝ, < ) ∈ (0..^𝑀) → sup({-𝑁, 𝑀}, ℝ, < ) < 𝑀)
131	129, 130	syl 14	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) < 𝑀)
132	81	nn0zd 9600	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℤ)
133	70	nn0zd 9600	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℤ)
134		zltnle 9525	. . . . . . . 8 ⊢ ((sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℤ ∧ 𝑀 ∈ ℤ) → (sup({-𝑁, 𝑀}, ℝ, < ) < 𝑀 ↔ ¬ 𝑀 ≤ sup({-𝑁, 𝑀}, ℝ, < )))
135	132, 133, 134	syl2anc 411	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (sup({-𝑁, 𝑀}, ℝ, < ) < 𝑀 ↔ ¬ 𝑀 ≤ sup({-𝑁, 𝑀}, ℝ, < )))
136	131, 135	mpbid 147	. . . . . 6 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 𝑀 ≤ sup({-𝑁, 𝑀}, ℝ, < ))
137	73, 136	pm2.65da 667	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ -𝑁 ∈ ℕ)
138	137	intnand 938	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ))
139		simpll 527	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℤ)
140		elznn0nn 9493	. . . . 5 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
141	139, 140	sylib 122	. . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (𝑁 ∈ ℕ₀ ∨ (𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ)))
142	138, 141	ecased 1385	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℕ₀)
143		simplr 529	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑀 ∈ ℕ₀)
144		simpr 110	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → (bits‘𝑁) ⊆ (0..^𝑀))
145		eqid 2231	. . 3 ⊢ inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < )
146	142, 143, 144, 145	bitsfzolem 12517	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ (0..^(2↑𝑀)))
147	67, 146	impbida 600	1 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) → (𝑁 ∈ (0..^(2↑𝑀)) ↔ (bits‘𝑁) ⊆ (0..^𝑀)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 DECID wdc 841 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 {crab 2514 ⊆ wss 3200 {cpr 3670 class class class wbr 4088 ‘cfv 5326 (class class class)co 6018 supcsup 7181 infcinf 7182 ℝcr 8031 0cc0 8032 1c1 8033 + caddc 8035 · cmul 8037 < clt 8214 ≤ cle 8215 -cneg 8351 / cdiv 8852 ℕcn 9143 2c2 9194 ℕ₀cn0 9402 ℤcz 9479 ℤ_≥cuz 9755 ℚcq 9853 ℝ⁺crp 9888 ..^cfzo 10377 ⌊cfl 10529 ↑cexp 10801 ∥ cdvds 12350 bitscbits 12503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-xor 1420 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fz 10244 df-fzo 10378 df-fl 10531 df-seqfrec 10711 df-exp 10802 df-cj 11404 df-re 11405 df-im 11406 df-rsqrt 11560 df-abs 11561 df-dvds 12351 df-bits 12504

This theorem is referenced by: bitsfi 12520 0bits 12522 bitsinv1 12525

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