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Theorem fsum0diaglem 11390
Description: Lemma for fisum0diag 11391. (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 9970 . . . . . . 7 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
21adantr 274 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 0 ≤ 𝑗)
3 elfz3nn0 10058 . . . . . . . . . 10 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
43adantr 274 . . . . . . . . 9 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
54nn0zd 9319 . . . . . . . 8 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℤ)
65zred 9321 . . . . . . 7 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℝ)
7 elfzelz 9968 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
87adantr 274 . . . . . . . 8 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
98zred 9321 . . . . . . 7 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℝ)
106, 9subge02d 8443 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (0 ≤ 𝑗 ↔ (𝑁𝑗) ≤ 𝑁))
112, 10mpbid 146 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ≤ 𝑁)
125, 8zsubcld 9326 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℤ)
13 eluz 9487 . . . . . 6 (((𝑁𝑗) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑁𝑗)) ↔ (𝑁𝑗) ≤ 𝑁))
1412, 5, 13syl2anc 409 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁 ∈ (ℤ‘(𝑁𝑗)) ↔ (𝑁𝑗) ≤ 𝑁))
1511, 14mpbird 166 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ (ℤ‘(𝑁𝑗)))
16 fzss2 10007 . . . 4 (𝑁 ∈ (ℤ‘(𝑁𝑗)) → (0...(𝑁𝑗)) ⊆ (0...𝑁))
1715, 16syl 14 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (0...(𝑁𝑗)) ⊆ (0...𝑁))
18 simpr 109 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ (0...(𝑁𝑗)))
1917, 18sseldd 3148 . 2 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ (0...𝑁))
20 elfzelz 9968 . . . . . 6 (𝑘 ∈ (0...(𝑁𝑗)) → 𝑘 ∈ ℤ)
2120adantl 275 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ ℤ)
2221zred 9321 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ ℝ)
23 elfzle2 9971 . . . . 5 (𝑘 ∈ (0...(𝑁𝑗)) → 𝑘 ≤ (𝑁𝑗))
2423adantl 275 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ≤ (𝑁𝑗))
2522, 6, 9, 24lesubd 8455 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ≤ (𝑁𝑘))
26 elfzuz 9964 . . . . 5 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ (ℤ‘0))
2726adantr 274 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ (ℤ‘0))
285, 21zsubcld 9326 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑘) ∈ ℤ)
29 elfz5 9960 . . . 4 ((𝑗 ∈ (ℤ‘0) ∧ (𝑁𝑘) ∈ ℤ) → (𝑗 ∈ (0...(𝑁𝑘)) ↔ 𝑗 ≤ (𝑁𝑘)))
3027, 28, 29syl2anc 409 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑗 ∈ (0...(𝑁𝑘)) ↔ 𝑗 ≤ (𝑁𝑘)))
3125, 30mpbird 166 . 2 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ (0...(𝑁𝑘)))
3219, 31jca 304 1 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁𝑘))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2141  wss 3121   class class class wbr 3987  cfv 5196  (class class class)co 5850  0cc0 7761  cle 7942  cmin 8077  0cn0 9122  cz 9199  cuz 9474  ...cfz 9952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-ltadd 7877
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-inn 8866  df-n0 9123  df-z 9200  df-uz 9475  df-fz 9953
This theorem is referenced by:  fisum0diag  11391  fprod0diagfz  11578
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