Theorem fsumfldivdiaglem 27126

Description: Lemma for fsumfldivdiag 27127. (Contributed by Mario Carneiro, 10-May-2016.)

Hypothesis

Ref	Expression
fsumfldivdiag.1	⊢ (𝜑 → 𝐴 ∈ ℝ)

Assertion

Ref	Expression
fsumfldivdiaglem	⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))))

Distinct variable groups:   𝑚,𝑛,𝐴   𝜑,𝑚,𝑛

Proof of Theorem fsumfldivdiaglem

Step	Hyp	Ref	Expression
1		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))
2		fsumfldivdiag.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ)
3	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℝ)
4		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘𝐴)))
5		fznnfl 13766	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝐴)))
6	3, 5	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝐴)))
7	4, 6	mpbid 232	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝐴))
8	7	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℕ)
9	3, 8	nndivred 12179	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ∈ ℝ)
10		fznnfl 13766	. . . . . . 7 ⊢ ((𝐴 / 𝑛) ∈ ℝ → (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛))))
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛))))
12	1, 11	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛)))
13	12	simpld 494	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ ℕ)
14	13	nnred 12140	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ ℝ)
15	12	simprd 495	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ≤ (𝐴 / 𝑛))
16	3	recnd 11140	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℂ)
17	16	mullidd 11130	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) = 𝐴)
18	8	nnge1d 12173	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ≤ 𝑛)
19		1red 11113	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ∈ ℝ)
20	8	nnred 12140	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℝ)
21		0red 11115	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 ∈ ℝ)
22	8, 13	nnmulcld 12178	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℕ)
23	22	nnred 12140	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℝ)
24	22	nngt0d 12174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < (𝑛 · 𝑚))
25	8	nngt0d 12174	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑛)
26		lemuldiv2 12003	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝑛 · 𝑚) ≤ 𝐴 ↔ 𝑚 ≤ (𝐴 / 𝑛)))
27	14, 3, 20, 25, 26	syl112anc 1376	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝑛 · 𝑚) ≤ 𝐴 ↔ 𝑚 ≤ (𝐴 / 𝑛)))
28	15, 27	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ≤ 𝐴)
29	21, 23, 3, 24, 28	ltletrd 11273	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝐴)
30		lemul1 11973	. . . . . . . . 9 ⊢ ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑛 ↔ (1 · 𝐴) ≤ (𝑛 · 𝐴)))
31	19, 20, 3, 29, 30	syl112anc 1376	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 ≤ 𝑛 ↔ (1 · 𝐴) ≤ (𝑛 · 𝐴)))
32	18, 31	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) ≤ (𝑛 · 𝐴))
33	17, 32	eqbrtrrd 5113	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ≤ (𝑛 · 𝐴))
34		ledivmul 11998	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝐴 / 𝑛) ≤ 𝐴 ↔ 𝐴 ≤ (𝑛 · 𝐴)))
35	3, 3, 20, 25, 34	syl112anc 1376	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝐴 / 𝑛) ≤ 𝐴 ↔ 𝐴 ≤ (𝑛 · 𝐴)))
36	33, 35	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ≤ 𝐴)
37	14, 9, 3, 15, 36	letrd 11270	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ≤ 𝐴)
38		fznnfl 13766	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ 𝐴)))
39	3, 38	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ 𝐴)))
40	13, 37, 39	mpbir2and 713	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ (1...(⌊‘𝐴)))
41	13	nngt0d 12174	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑚)
42		lemuldiv 12002	. . . . . 6 ⊢ ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → ((𝑛 · 𝑚) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑚)))
43	20, 3, 14, 41, 42	syl112anc 1376	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝑛 · 𝑚) ≤ 𝐴 ↔ 𝑛 ≤ (𝐴 / 𝑚)))
44	28, 43	mpbid 232	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ≤ (𝐴 / 𝑚))
45	3, 13	nndivred 12179	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑚) ∈ ℝ)
46		fznnfl 13766	. . . . 5 ⊢ ((𝐴 / 𝑚) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑚))))
47	45, 46	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑚))))
48	8, 44, 47	mpbir2and 713	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))
49	40, 48	jca 511	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚)))))
50	49	ex 412	1 ⊢ (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2111   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346  ℝcr 11005  0cc0 11006  1c1 11007   · cmul 11011   < clt 11146   ≤ cle 11147   / cdiv 11774  ℕcn 12125  ...cfz 13407  ⌊cfl 13694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fl 13696

This theorem is referenced by:  fsumfldivdiag  27127