Theorem fsumfldivdiaglem 27232

Description: Lemma for fsumfldivdiag 27233. (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 773	. . . . . 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 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘𝐴)))
5		fznnfl 13902	. . . . . . . . . . 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 12320	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ∈ ℝ)
10		fznnfl 13902	. . . . . . 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 12281	. . . . 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 11289	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℂ)
17	16	mullidd 11279	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) = 𝐴)
18	8	nnge1d 12314	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ≤ 𝑛)
19		1red 11262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ∈ ℝ)
20	8	nnred 12281	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℝ)
21		0red 11264	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 ∈ ℝ)
22	8, 13	nnmulcld 12319	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℕ)
23	22	nnred 12281	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℝ)
24	22	nngt0d 12315	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < (𝑛 · 𝑚))
25	8	nngt0d 12315	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑛)
26		lemuldiv2 12149	. . . . . . . . . . . 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 11421	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝐴)
30		lemul1 12119	. . . . . . . . 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 5167	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ≤ (𝑛 · 𝐴))
34		ledivmul 12144	. . . . . . 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 11418	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ≤ 𝐴)
38		fznnfl 13902	. . . . 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 12315	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑚)
42		lemuldiv 12148	. . . . . 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 12320	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑚) ∈ ℝ)
46		fznnfl 13902	. . . . 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 2108   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295   ≤ cle 11296   / cdiv 11920  ℕcn 12266  ...cfz 13547  ⌊cfl 13830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fl 13832

This theorem is referenced by:  fsumfldivdiag  27233