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Theorem fsumfldivdiaglem 26327
Description: Lemma for fsumfldivdiag 26328. (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 770 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))
2 fsumfldivdiag.1 . . . . . . . . 9 (𝜑𝐴 ∈ ℝ)
32adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℝ)
4 simprl 768 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘𝐴)))
5 fznnfl 13571 . . . . . . . . . . 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 231 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 ∈ ℕ ∧ 𝑛𝐴))
87simpld 495 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℕ)
93, 8nndivred 12016 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ∈ ℝ)
10 fznnfl 13571 . . . . . . 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 231 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ ℕ ∧ 𝑚 ≤ (𝐴 / 𝑛)))
1312simpld 495 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ ℕ)
1413nnred 11977 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ ℝ)
1512simprd 496 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ≤ (𝐴 / 𝑛))
163recnd 10992 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℂ)
1716mulid2d 10982 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) = 𝐴)
188nnge1d 12010 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ≤ 𝑛)
19 1red 10965 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ∈ ℝ)
208nnred 11977 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℝ)
21 0red 10967 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 ∈ ℝ)
228, 13nnmulcld 12015 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℕ)
2322nnred 11977 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℝ)
2422nngt0d 12011 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < (𝑛 · 𝑚))
258nngt0d 12011 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑛)
26 lemuldiv2 11845 . . . . . . . . . . . 12 ((𝑚 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝑛 · 𝑚) ≤ 𝐴𝑚 ≤ (𝐴 / 𝑛)))
2714, 3, 20, 25, 26syl112anc 1373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝑛 · 𝑚) ≤ 𝐴𝑚 ≤ (𝐴 / 𝑛)))
2815, 27mpbird 256 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ≤ 𝐴)
2921, 23, 3, 24, 28ltletrd 11124 . . . . . . . . 9 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝐴)
30 lemul1 11816 . . . . . . . . 9 ((1 ∈ ℝ ∧ 𝑛 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (1 ≤ 𝑛 ↔ (1 · 𝐴) ≤ (𝑛 · 𝐴)))
3119, 20, 3, 29, 30syl112anc 1373 . . . . . . . 8 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 ≤ 𝑛 ↔ (1 · 𝐴) ≤ (𝑛 · 𝐴)))
3218, 31mpbid 231 . . . . . . 7 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) ≤ (𝑛 · 𝐴))
3317, 32eqbrtrrd 5099 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ≤ (𝑛 · 𝐴))
34 ledivmul 11840 . . . . . . 7 ((𝐴 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑛 ∈ ℝ ∧ 0 < 𝑛)) → ((𝐴 / 𝑛) ≤ 𝐴𝐴 ≤ (𝑛 · 𝐴)))
353, 3, 20, 25, 34syl112anc 1373 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝐴 / 𝑛) ≤ 𝐴𝐴 ≤ (𝑛 · 𝐴)))
3633, 35mpbird 256 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ≤ 𝐴)
3714, 9, 3, 15, 36letrd 11121 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚𝐴)
38 fznnfl 13571 . . . . 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 710 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ∈ (1...(⌊‘𝐴)))
4113nngt0d 12011 . . . . . 6 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑚)
42 lemuldiv 11844 . . . . . 6 ((𝑛 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝑚 ∈ ℝ ∧ 0 < 𝑚)) → ((𝑛 · 𝑚) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑚)))
4320, 3, 14, 41, 42syl112anc 1373 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → ((𝑛 · 𝑚) ≤ 𝐴𝑛 ≤ (𝐴 / 𝑚)))
4428, 43mpbid 231 . . . 4 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ≤ (𝐴 / 𝑚))
453, 13nndivred 12016 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑚) ∈ ℝ)
46 fznnfl 13571 . . . . 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 710 . . 3 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚))))
4940, 48jca 512 . 2 ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑚 ∈ (1...(⌊‘𝐴)) ∧ 𝑛 ∈ (1...(⌊‘(𝐴 / 𝑚)))))
5049ex 413 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 205  wa 396  wcel 2106   class class class wbr 5075  cfv 6428  (class class class)co 7269  cr 10859  0cc0 10860  1c1 10861   · cmul 10865   < clt 10998  cle 10999   / cdiv 11621  cn 11962  ...cfz 13228  cfl 13499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7580  ax-cnex 10916  ax-resscn 10917  ax-1cn 10918  ax-icn 10919  ax-addcl 10920  ax-addrcl 10921  ax-mulcl 10922  ax-mulrcl 10923  ax-mulcom 10924  ax-addass 10925  ax-mulass 10926  ax-distr 10927  ax-i2m1 10928  ax-1ne0 10929  ax-1rid 10930  ax-rnegex 10931  ax-rrecex 10932  ax-cnre 10933  ax-pre-lttri 10934  ax-pre-lttrn 10935  ax-pre-ltadd 10936  ax-pre-mulgt0 10937  ax-pre-sup 10938
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5486  df-eprel 5492  df-po 5500  df-so 5501  df-fr 5541  df-we 5543  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-rn 5597  df-res 5598  df-ima 5599  df-pred 6197  df-ord 6264  df-on 6265  df-lim 6266  df-suc 6267  df-iota 6386  df-fun 6430  df-fn 6431  df-f 6432  df-f1 6433  df-fo 6434  df-f1o 6435  df-fv 6436  df-riota 7226  df-ov 7272  df-oprab 7273  df-mpo 7274  df-om 7705  df-1st 7822  df-2nd 7823  df-frecs 8086  df-wrecs 8117  df-recs 8191  df-rdg 8230  df-er 8487  df-en 8723  df-dom 8724  df-sdom 8725  df-sup 9190  df-inf 9191  df-pnf 11000  df-mnf 11001  df-xr 11002  df-ltxr 11003  df-le 11004  df-sub 11196  df-neg 11197  df-div 11622  df-nn 11963  df-n0 12223  df-z 12309  df-uz 12572  df-fz 13229  df-fl 13501
This theorem is referenced by:  fsumfldivdiag  26328
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