modqmulnn - Intuitionistic Logic Explorer

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Theorem modqmulnn 10453

Description: Move a positive integer in and out of a floor in the first argument of a modulo operation. (Contributed by Jim Kingdon, 18-Oct-2021.)

**Assertion**
Ref	Expression
modqmulnn	⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))

**Proof of Theorem modqmulnn**
Step	Hyp	Ref	Expression
1		nnq 9726	. . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℚ)
2	1	3ad2ant1 1020	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑁 ∈ ℚ)
3		flqcl 10382	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℤ)
4		zq 9719	. . . . . . 7 ⊢ ((⌊‘𝐴) ∈ ℤ → (⌊‘𝐴) ∈ ℚ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℚ)
6	5	3ad2ant2 1021	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘𝐴) ∈ ℚ)
7		qmulcl 9730	. . . . 5 ⊢ ((𝑁 ∈ ℚ ∧ (⌊‘𝐴) ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ∈ ℚ)
8	2, 6, 7	syl2anc 411	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · (⌊‘𝐴)) ∈ ℚ)
9		qre 9718	. . . 4 ⊢ ((𝑁 · (⌊‘𝐴)) ∈ ℚ → (𝑁 · (⌊‘𝐴)) ∈ ℝ)
10	8, 9	syl 14	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · (⌊‘𝐴)) ∈ ℝ)
11		simp2 1000	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝐴 ∈ ℚ)
12		qmulcl 9730	. . . . . 6 ⊢ ((𝑁 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (𝑁 · 𝐴) ∈ ℚ)
13	2, 11, 12	syl2anc 411	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝐴) ∈ ℚ)
14	13	flqcld 10386	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘(𝑁 · 𝐴)) ∈ ℤ)
15	14	zred 9467	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘(𝑁 · 𝐴)) ∈ ℝ)
16		nnmulcl 9030	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) ∈ ℕ)
17		nnq 9726	. . . . . . 7 ⊢ ((𝑁 · 𝑀) ∈ ℕ → (𝑁 · 𝑀) ∈ ℚ)
18	16, 17	syl 14	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) ∈ ℚ)
19	18	3adant2 1018	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) ∈ ℚ)
20		qre 9718	. . . . 5 ⊢ ((𝑁 · 𝑀) ∈ ℚ → (𝑁 · 𝑀) ∈ ℝ)
21	19, 20	syl 14	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) ∈ ℝ)
22		simp1 999	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑁 ∈ ℕ)
23	22	nncnd 9023	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑁 ∈ ℂ)
24		simp3 1001	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℕ)
25	24	nncnd 9023	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℂ)
26	22	nnap0d 9055	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑁 # 0)
27	24	nnap0d 9055	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑀 # 0)
28	23, 25, 26, 27	mulap0d 8704	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) # 0)
29		0z 9356	. . . . . . . . . 10 ⊢ 0 ∈ ℤ
30		zq 9719	. . . . . . . . . 10 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
31	29, 30	ax-mp 5	. . . . . . . . 9 ⊢ 0 ∈ ℚ
32		qapne 9732	. . . . . . . . 9 ⊢ (((𝑁 · 𝑀) ∈ ℚ ∧ 0 ∈ ℚ) → ((𝑁 · 𝑀) # 0 ↔ (𝑁 · 𝑀) ≠ 0))
33	19, 31, 32	sylancl 413	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · 𝑀) # 0 ↔ (𝑁 · 𝑀) ≠ 0))
34	28, 33	mpbid 147	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) ≠ 0)
35		qdivcl 9736	. . . . . . 7 ⊢ (((𝑁 · (⌊‘𝐴)) ∈ ℚ ∧ (𝑁 · 𝑀) ∈ ℚ ∧ (𝑁 · 𝑀) ≠ 0) → ((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀)) ∈ ℚ)
36	8, 19, 34, 35	syl3anc 1249	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀)) ∈ ℚ)
37	36	flqcld 10386	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))) ∈ ℤ)
38	37	zred 9467	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))) ∈ ℝ)
39	21, 38	remulcld 8076	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀)))) ∈ ℝ)
40		nnnn0 9275	. . . . 5 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
41		flqmulnn0 10408	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
42	40, 41	sylan 283	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
43	22, 11, 42	syl2anc 411	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · (⌊‘𝐴)) ≤ (⌊‘(𝑁 · 𝐴)))
44	10, 15, 39, 43	lesub1dd 8607	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) − ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))) ≤ ((⌊‘(𝑁 · 𝐴)) − ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))))
45	22	nnred 9022	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑁 ∈ ℝ)
46	24	nnred 9022	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℝ)
47	22	nngt0d 9053	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 0 < 𝑁)
48	24	nngt0d 9053	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 0 < 𝑀)
49	45, 46, 47, 48	mulgt0d 8168	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 0 < (𝑁 · 𝑀))
50		modqval 10435	. . 3 ⊢ (((𝑁 · (⌊‘𝐴)) ∈ ℚ ∧ (𝑁 · 𝑀) ∈ ℚ ∧ 0 < (𝑁 · 𝑀)) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) = ((𝑁 · (⌊‘𝐴)) − ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))))
51	8, 19, 49, 50	syl3anc 1249	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) = ((𝑁 · (⌊‘𝐴)) − ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))))
52		zq 9719	. . . . 5 ⊢ ((⌊‘(𝑁 · 𝐴)) ∈ ℤ → (⌊‘(𝑁 · 𝐴)) ∈ ℚ)
53	14, 52	syl 14	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘(𝑁 · 𝐴)) ∈ ℚ)
54		modqval 10435	. . . 4 ⊢ (((⌊‘(𝑁 · 𝐴)) ∈ ℚ ∧ (𝑁 · 𝑀) ∈ ℚ ∧ 0 < (𝑁 · 𝑀)) → ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)) = ((⌊‘(𝑁 · 𝐴)) − ((𝑁 · 𝑀) · (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀))))))
55	53, 19, 49, 54	syl3anc 1249	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)) = ((⌊‘(𝑁 · 𝐴)) − ((𝑁 · 𝑀) · (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀))))))
56	16	3adant2 1018	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (𝑁 · 𝑀) ∈ ℕ)
57		flqdiv 10432	. . . . . . 7 ⊢ (((𝑁 · 𝐴) ∈ ℚ ∧ (𝑁 · 𝑀) ∈ ℕ) → (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀))) = (⌊‘((𝑁 · 𝐴) / (𝑁 · 𝑀))))
58	13, 56, 57	syl2anc 411	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀))) = (⌊‘((𝑁 · 𝐴) / (𝑁 · 𝑀))))
59		flqdiv 10432	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑀)) = (⌊‘(𝐴 / 𝑀)))
60	59	3adant1 1017	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((⌊‘𝐴) / 𝑀)) = (⌊‘(𝐴 / 𝑀)))
61	3	zcnd 9468	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (⌊‘𝐴) ∈ ℂ)
62	11, 61	syl 14	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘𝐴) ∈ ℂ)
63	62, 25, 23, 27, 26	divcanap5d 8863	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀)) = ((⌊‘𝐴) / 𝑀))
64	63	fveq2d 5565	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))) = (⌊‘((⌊‘𝐴) / 𝑀)))
65		qcn 9727	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
66	11, 65	syl 14	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → 𝐴 ∈ ℂ)
67	66, 25, 23, 27, 26	divcanap5d 8863	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · 𝐴) / (𝑁 · 𝑀)) = (𝐴 / 𝑀))
68	67	fveq2d 5565	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((𝑁 · 𝐴) / (𝑁 · 𝑀))) = (⌊‘(𝐴 / 𝑀)))
69	60, 64, 68	3eqtr4rd 2240	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((𝑁 · 𝐴) / (𝑁 · 𝑀))) = (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))
70	58, 69	eqtrd 2229	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀))) = (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))
71	70	oveq2d 5941	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · 𝑀) · (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀)))) = ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀)))))
72	71	oveq2d 5941	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((⌊‘(𝑁 · 𝐴)) − ((𝑁 · 𝑀) · (⌊‘((⌊‘(𝑁 · 𝐴)) / (𝑁 · 𝑀))))) = ((⌊‘(𝑁 · 𝐴)) − ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))))
73	55, 72	eqtrd 2229	. 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)) = ((⌊‘(𝑁 · 𝐴)) − ((𝑁 · 𝑀) · (⌊‘((𝑁 · (⌊‘𝐴)) / (𝑁 · 𝑀))))))
74	44, 51, 73	3brtr4d 4066	1 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ ℚ ∧ 𝑀 ∈ ℕ) → ((𝑁 · (⌊‘𝐴)) mod (𝑁 · 𝑀)) ≤ ((⌊‘(𝑁 · 𝐴)) mod (𝑁 · 𝑀)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2167 ≠ wne 2367 class class class wbr 4034 ‘cfv 5259 (class class class)co 5925 ℂcc 7896 ℝcr 7897 0cc0 7898 · cmul 7903 < clt 8080 ≤ cle 8081 − cmin 8216 # cap 8627 / cdiv 8718 ℕcn 9009 ℕ₀cn0 9268 ℤcz 9345 ℚcq 9712 ⌊cfl 10377 mod cmo 10433

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-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 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-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-po 4332 df-iso 4333 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-n0 9269 df-z 9346 df-q 9713 df-rp 9748 df-fl 10379 df-mod 10434

This theorem is referenced by: (None)

W3C validator