bitsfzo - Intuitionistic Logic Explorer

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Theorem bitsfzo 12119

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 12108	. . . 4 ⊢ (𝑚 ∈ (bits‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚)))))
2		simp32 1036	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℕ₀)
3		nn0uz 9636	. . . . . . 7 ⊢ ℕ₀ = (ℤ_≥‘0)
4	2, 3	eleqtrdi 2289	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (ℤ_≥‘0))
5		simp1r 1024	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℕ₀)
6	5	nn0zd 9446	. . . . . 6 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑀 ∈ ℤ)
7		2re 9060	. . . . . . . . . 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 10782	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ)
10		simp1l 1023	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℤ)
11	10	zred 9448	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑁 ∈ ℝ)
12	8, 5	reexpcld 10782	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑀) ∈ ℝ)
13	9	recnd 8055	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℂ)
14	13	mullidd 8044	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (1 · (2↑𝑚)) = (2↑𝑚))
15		1z 9352	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℤ
16		zq 9700	. . . . . . . . . . . . . 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 9152	. . . . . . . . . . . . . . 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 10787	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℕ)
22		znq 9698	. . . . . . . . . . . . 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 10336	. . . . . . . . . . . 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 1037	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))
27		qltnle 10333	. . . . . . . . . . . . . 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 9104	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
30	29	breq2i 4041	. . . . . . . . . . . . . 14 ⊢ ((𝑁 / (2↑𝑚)) < (0 + 1) ↔ (𝑁 / (2↑𝑚)) < 1)
31		2rp 9733	. . . . . . . . . . . . . . . . . 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 9446	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ ℤ)
34	32, 33	rpexpcld 10789	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ∈ ℝ⁺)
35		elfzole1 10231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 0 ≤ 𝑁)
36	35	3ad2ant2 1021	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ 𝑁)
37	11, 34, 36	divge0d 9812	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 0 ≤ (𝑁 / (2↑𝑚)))
38		0z 9337	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ ℤ
39		flqbi 10380	. . . . . . . . . . . . . . . . 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 12076	. . . . . . . . . . . . . . . . 17 ⊢ 2 ∥ 0
42		id 19	. . . . . . . . . . . . . . . . 17 ⊢ ((⌊‘(𝑁 / (2↑𝑚))) = 0 → (⌊‘(𝑁 / (2↑𝑚))) = 0)
43	41, 42	breqtrrid 4071	. . . . . . . . . . . . . . . 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 664	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → ¬ ¬ 1 ≤ (𝑁 / (2↑𝑚)))
49		notnotrdc 844	. . . . . . . . . . 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 8041	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 1 ∈ ℝ)
52	51, 11, 34	lemuldivd 9821	. . . . . . . . . 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 4057	. . . . . . . 8 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) ≤ 𝑁)
55		elfzolt2 10232	. . . . . . . . 9 ⊢ (𝑁 ∈ (0..^(2↑𝑀)) → 𝑁 < (2↑𝑀))
56	55	3ad2ant2 1021	. . . . . . . 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 8151	. . . . . . 7 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → (2↑𝑚) < (2↑𝑀))
58		1lt2 9160	. . . . . . . . 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 10801	. . . . . . . 8 ⊢ (((2 ∈ ℝ ∧ 𝑚 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀) ∧ 1 < 2) → (𝑚 < 𝑀 ↔ (2↑𝑚) < (2↑𝑀)))
61	8, 2, 5, 59, 60	syl31anc 1252	. . . . . . 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 10225	. . . . . 6 ⊢ (𝑚 ∈ (0..^𝑀) ↔ (𝑚 ∈ (ℤ_≥‘0) ∧ 𝑀 ∈ ℤ ∧ 𝑚 < 𝑀))
64	4, 6, 62, 63	syl3anbrc 1183	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ 𝑁 ∈ (0..^(2↑𝑀)) ∧ (𝑁 ∈ ℤ ∧ 𝑚 ∈ ℕ₀ ∧ ¬ 2 ∥ (⌊‘(𝑁 / (2↑𝑚))))) → 𝑚 ∈ (0..^𝑀))
65	64	3expia 1207	. . . 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 3189	. 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 9003	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℝ)
70		simpllr 534	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℕ₀)
71	70	nn0red 9303	. . . . . . 7 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℝ)
72		maxle2 11377	. . . . . . 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 528	. . . . . . . . 9 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (bits‘𝑁) ⊆ (0..^𝑀))
75		n2dvdsm1 12078	. . . . . . . . . . 11 ⊢ ¬ 2 ∥ -1
76		simplll 533	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
77	76	zred 9448	. . . . . . . . . . . . . . 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 9302	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ₀)
80		nn0maxcl 11390	. . . . . . . . . . . . . . . . 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 10787	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℕ)
83	77, 82	nndivred 9040	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ∈ ℝ)
84		1red 8041	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 1 ∈ ℝ)
85	76	zcnd 9449	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ)
86	82	nncnd 9004	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℂ)
87	82	nnap0d 9036	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) # 0)
88	85, 86, 87	divnegapd 8830	. . . . . . . . . . . . . . 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 9303	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℝ)
90	82	nnred 9003	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℝ)
91		maxle1 11376	. . . . . . . . . . . . . . . . . . 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 9354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 2 ∈ ℤ
94		uzid 9615	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
95	93, 94	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 2 ∈ (ℤ_≥‘2)
96		bernneq3 10754	. . . . . . . . . . . . . . . . . . . 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 8145	. . . . . . . . . . . . . . . . . 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 8150	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ (2↑sup({-𝑁, 𝑀}, ℝ, < )))
100	86	mulridd 8043	. . . . . . . . . . . . . . . . 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 4061	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -𝑁 ≤ ((2↑sup({-𝑁, 𝑀}, ℝ, < )) · 1))
102	82	nnrpd 9769	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2↑sup({-𝑁, 𝑀}, ℝ, < )) ∈ ℝ⁺)
103	69, 84, 102	ledivmuld 9825	. . . . . . . . . . . . . . . 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 4055	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) ≤ 1)
106	83, 84, 105	lenegcon1d 8554	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → -1 ≤ (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
107	68	nngt0d 9034	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -𝑁)
108	82	nngt0d 9034	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (2↑sup({-𝑁, 𝑀}, ℝ, < )))
109	69, 90, 107, 108	divgt0d 8962	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < (-𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
110	109, 88	breqtrrd 4061	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 0 < -(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))))
111	83	lt0neg1d 8542	. . . . . . . . . . . . . . 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 7972	. . . . . . . . . . . . . . 15 ⊢ 1 ∈ ℂ
114		neg1cn 9095	. . . . . . . . . . . . . . 15 ⊢ -1 ∈ ℂ
115		1pneg1e0 9101	. . . . . . . . . . . . . . 15 ⊢ (1 + -1) = 0
116	113, 114, 115	addcomli 8171	. . . . . . . . . . . . . 14 ⊢ (-1 + 1) = 0
117	112, 116	breqtrrdi 4075	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < ))) < (-1 + 1))
118		znq 9698	. . . . . . . . . . . . . . 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 9358	. . . . . . . . . . . . . 14 ⊢ -1 ∈ ℤ
121		flqbi 10380	. . . . . . . . . . . . . 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 946	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))) = -1)
124	123	breq2d 4045	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → (2 ∥ (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))) ↔ 2 ∥ -1))
125	75, 124	mtbiri 676	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → ¬ 2 ∥ (⌊‘(𝑁 / (2↑sup({-𝑁, 𝑀}, ℝ, < )))))
126		bitsval2 12109	. . . . . . . . . . 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 3184	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ (0..^𝑀))
130		elfzolt2 10232	. . . . . . . 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 9446	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → sup({-𝑁, 𝑀}, ℝ, < ) ∈ ℤ)
133	70	nn0zd 9446	. . . . . . . 8 ⊢ ((((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) ∧ -𝑁 ∈ ℕ) → 𝑀 ∈ ℤ)
134		zltnle 9372	. . . . . . . 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 662	. . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → ¬ -𝑁 ∈ ℕ)
138	137	intnand 932	. . . 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 9340	. . . . 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 1360	. . 3 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ ℕ₀)
143		simplr 528	. . 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 2196	. . 3 ⊢ inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < ) = inf({𝑛 ∈ ℕ₀ ∣ 𝑁 < (2↑𝑛)}, ℝ, < )
146	142, 143, 144, 145	bitsfzolem 12118	. 2 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℕ₀) ∧ (bits‘𝑁) ⊆ (0..^𝑀)) → 𝑁 ∈ (0..^(2↑𝑀)))
147	67, 146	impbida 596	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 709 DECID wdc 835 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 {crab 2479 ⊆ wss 3157 {cpr 3623 class class class wbr 4033 ‘cfv 5258 (class class class)co 5922 supcsup 7048 infcinf 7049 ℝcr 7878 0cc0 7879 1c1 7880 + caddc 7882 · cmul 7884 < clt 8061 ≤ cle 8062 -cneg 8198 / cdiv 8699 ℕcn 8990 2c2 9041 ℕ₀cn0 9249 ℤcz 9326 ℤ_≥cuz 9601 ℚcq 9693 ℝ⁺crp 9728 ..^cfzo 10217 ⌊cfl 10358 ↑cexp 10630 ∥ cdvds 11952 bitscbits 12105

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-bits 12106

This theorem is referenced by: (None)

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