Theorem fsumfldivdiaglem 27156

Description: Lemma for fsumfldivdiag 27157. (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 13884	. . . . . . . . . . 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 12299	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑛) ∈ ℝ)
10		fznnfl 13884	. . . . . . 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 12260	. . . . 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 11268	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ∈ ℂ)
17	16	mullidd 11258	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (1 · 𝐴) = 𝐴)
18	8	nnge1d 12293	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ≤ 𝑛)
19		1red 11241	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 1 ∈ ℝ)
20	8	nnred 12260	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑛 ∈ ℝ)
21		0red 11243	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 ∈ ℝ)
22	8, 13	nnmulcld 12298	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℕ)
23	22	nnred 12260	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝑛 · 𝑚) ∈ ℝ)
24	22	nngt0d 12294	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < (𝑛 · 𝑚))
25	8	nngt0d 12294	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑛)
26		lemuldiv2 12128	. . . . . . . . . . . 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 11400	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝐴)
30		lemul1 12098	. . . . . . . . 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 5148	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝐴 ≤ (𝑛 · 𝐴))
34		ledivmul 12123	. . . . . . 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 11397	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 𝑚 ≤ 𝐴)
38		fznnfl 13884	. . . . 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 12294	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → 0 < 𝑚)
42		lemuldiv 12127	. . . . . 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 12299	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (1...(⌊‘𝐴)) ∧ 𝑚 ∈ (1...(⌊‘(𝐴 / 𝑛))))) → (𝐴 / 𝑚) ∈ ℝ)
46		fznnfl 13884	. . . . 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 2109   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  ℝcr 11133  0cc0 11134  1c1 11135   · cmul 11139   < clt 11274   ≤ cle 11275   / cdiv 11899  ℕcn 12245  ...cfz 13529  ⌊cfl 13812

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9459  df-inf 9460  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-n0 12507  df-z 12594  df-uz 12858  df-fz 13530  df-fl 13814

This theorem is referenced by:  fsumfldivdiag  27157