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Theorem fsum0diaglem 14350
 Description: Lemma for fsum0diag 14351. (Contributed by Mario Carneiro, 28-Apr-2014.) (Revised by Mario Carneiro, 8-Apr-2016.)
Assertion
Ref Expression
fsum0diaglem ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁𝑘))))
Distinct variable group:   𝑗,𝑘,𝑁

Proof of Theorem fsum0diaglem
StepHypRef Expression
1 elfzle1 12215 . . . . . . 7 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
21adantr 480 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 0 ≤ 𝑗)
3 elfz3nn0 12303 . . . . . . . . . 10 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
43adantr 480 . . . . . . . . 9 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
54nn0zd 11356 . . . . . . . 8 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℤ)
65zred 11358 . . . . . . 7 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℝ)
7 elfzelz 12213 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
87adantr 480 . . . . . . . 8 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
98zred 11358 . . . . . . 7 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℝ)
106, 9subge02d 10498 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (0 ≤ 𝑗 ↔ (𝑁𝑗) ≤ 𝑁))
112, 10mpbid 221 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ≤ 𝑁)
125, 8zsubcld 11363 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℤ)
13 eluz 11577 . . . . . 6 (((𝑁𝑗) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑁𝑗)) ↔ (𝑁𝑗) ≤ 𝑁))
1412, 5, 13syl2anc 691 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁 ∈ (ℤ‘(𝑁𝑗)) ↔ (𝑁𝑗) ≤ 𝑁))
1511, 14mpbird 246 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ (ℤ‘(𝑁𝑗)))
16 fzss2 12252 . . . 4 (𝑁 ∈ (ℤ‘(𝑁𝑗)) → (0...(𝑁𝑗)) ⊆ (0...𝑁))
1715, 16syl 17 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (0...(𝑁𝑗)) ⊆ (0...𝑁))
18 simpr 476 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ (0...(𝑁𝑗)))
1917, 18sseldd 3569 . 2 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ (0...𝑁))
20 elfzelz 12213 . . . . . 6 (𝑘 ∈ (0...(𝑁𝑗)) → 𝑘 ∈ ℤ)
2120adantl 481 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ ℤ)
2221zred 11358 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ ℝ)
23 elfzle2 12216 . . . . 5 (𝑘 ∈ (0...(𝑁𝑗)) → 𝑘 ≤ (𝑁𝑗))
2423adantl 481 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ≤ (𝑁𝑗))
2522, 6, 9, 24lesubd 10510 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ≤ (𝑁𝑘))
26 elfzuz 12209 . . . . 5 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ (ℤ‘0))
2726adantr 480 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ (ℤ‘0))
285, 21zsubcld 11363 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑘) ∈ ℤ)
29 elfz5 12205 . . . 4 ((𝑗 ∈ (ℤ‘0) ∧ (𝑁𝑘) ∈ ℤ) → (𝑗 ∈ (0...(𝑁𝑘)) ↔ 𝑗 ≤ (𝑁𝑘)))
3027, 28, 29syl2anc 691 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑗 ∈ (0...(𝑁𝑘)) ↔ 𝑗 ≤ (𝑁𝑘)))
3125, 30mpbird 246 . 2 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ (0...(𝑁𝑘)))
3219, 31jca 553 1 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁𝑘))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815   ≤ cle 9954   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198 This theorem is referenced by:  fsum0diag  14351  fprod0diag  14556
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