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Theorem fsumfldivdiaglem 24960
Description: Lemma for fsumfldivdiag 24961. (Contributed by Mario Carneiro, 10-May-2016.)
Hypothesis
Ref Expression
fsumfldivdiag.1 (𝜑𝐴 ∈ 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
StepHypRef Expression
1 simprr 811 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))
2 fsumfldivdiag.1 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
32adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℝ)
4 simprl 809 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘𝐴)))
5 fznnfl 12701 . . . . . . . . . . 11 (𝐴 ∈ ℝ → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
63, 5syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ (1...(⌊‘𝐴)) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝐴)))
74, 6mpbid 222 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ ℕ ∧ 𝑛𝐴))
87simpld 474 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℕ)
93, 8nndivred 11107 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ∈ ℝ)
10 fznnfl 12701 . . . . . . 7 ((𝐴 / 𝑛) ∈ ℝ → (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛))))
119, 10syl 17 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))) ↔ (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛))))
121, 11mpbid 222 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛)))
1312simpld 474 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ ℕ)
1413nnred 11073 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ ℝ)
1512simprd 478 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ≤ (𝐴 / 𝑛))
163recnd 10106 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℂ)
1716mulid2d 10096 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) = 𝐴)
188nnge1d 11101 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ≤ 𝑛)
19 1red 10093 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ∈ ℝ)
208nnred 11073 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℝ)
21 0red 10079 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 ∈ ℝ)
228, 13nnmulcld 11106 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℕ)
2322nnred 11073 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℝ)
2422nngt0d 11102 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < (𝑛 · 𝑚))
258nngt0d 11102 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑛)
26 lemuldiv2 10942 . . . . . . . . . . . 12 ((𝑚 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝑛 · 𝑚) ≤ 𝐴𝑚 ≤ (𝐴 / 𝑛)))
2714, 3, 20, 25, 26syl112anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝑛 · 𝑚) ≤ 𝐴𝑚 ≤ (𝐴 / 𝑛)))
2815, 27mpbird 247 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ≤ 𝐴)
2921, 23, 3, 24, 28ltletrd 10235 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝐴)
30 lemul1 10913 . . . . . . . . 9 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑛 ↔ (1 · 𝐴) ≤ (𝑛 · 𝐴)))
3119, 20, 3, 29, 30syl112anc 1370 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 ≤ 𝑛 ↔ (1 · 𝐴) ≤ (𝑛 · 𝐴)))
3218, 31mpbid 222 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) ≤ (𝑛 · 𝐴))
3317, 32eqbrtrrd 4709 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ≤ (𝑛 · 𝐴))
34 ledivmul 10937 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝐴 / 𝑛) ≤ 𝐴𝐴 ≤ (𝑛 · 𝐴)))
353, 3, 20, 25, 34syl112anc 1370 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝐴 / 𝑛) ≤ 𝐴𝐴 ≤ (𝑛 · 𝐴)))
3633, 35mpbird 247 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ≤ 𝐴)
3714, 9, 3, 15, 36letrd 10232 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚𝐴)
38 fznnfl 12701 . . . . 5 (𝐴 ∈ ℝ → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
393, 38syl 17 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘𝐴)) ↔ (𝑚 ∈ ℕ ∧ 𝑚𝐴)))
4013, 37, 39mpbir2and 977 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ (1...(⌊‘𝐴)))
4113nngt0d 11102 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑚)
42 lemuldiv 10941 . . . . . 6 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → ((𝑛 · 𝑚) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑚)))
4320, 3, 14, 41, 42syl112anc 1370 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝑛 · 𝑚) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑚)))
4428, 43mpbid 222 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ≤ (𝐴 / 𝑚))
453, 13nndivred 11107 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑚) ∈ ℝ)
46 fznnfl 12701 . . . . 5 ((𝐴 / 𝑚) ∈ ℝ → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑚))))
4745, 46syl 17 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ (𝐴 / 𝑚))))
488, 44, 47mpbir2and 977 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))
4940, 48jca 553 . 2 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚)))))
5049ex 449 1 (𝜑 → ((𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛)))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   · cmul 9979   < clt 10112  cle 10113   / cdiv 10722  cn 11058  ...cfz 12364  cfl 12631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fl 12633
This theorem is referenced by:  fsumfldivdiag  24961
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