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Theorem fsum0diaglem 14628
Description: Lemma for fsum0diag 14629. (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 12458 . . . . . . 7 (𝑗 ∈ (0...𝑁) → 0 ≤ 𝑗)
21adantr 472 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 0 ≤ 𝑗)
3 elfz3nn0 12548 . . . . . . . . . 10 (𝑗 ∈ (0...𝑁) → 𝑁 ∈ ℕ0)
43adantr 472 . . . . . . . . 9 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℕ0)
54nn0zd 11593 . . . . . . . 8 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℤ)
65zred 11595 . . . . . . 7 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ ℝ)
7 elfzelz 12456 . . . . . . . . 9 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ ℤ)
87adantr 472 . . . . . . . 8 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℤ)
98zred 11595 . . . . . . 7 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ ℝ)
106, 9subge02d 10732 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (0 ≤ 𝑗 ↔ (𝑁𝑗) ≤ 𝑁))
112, 10mpbid 222 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ≤ 𝑁)
125, 8zsubcld 11600 . . . . . 6 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑗) ∈ ℤ)
13 eluz 11814 . . . . . 6 (((𝑁𝑗) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ‘(𝑁𝑗)) ↔ (𝑁𝑗) ≤ 𝑁))
1412, 5, 13syl2anc 696 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁 ∈ (ℤ‘(𝑁𝑗)) ↔ (𝑁𝑗) ≤ 𝑁))
1511, 14mpbird 247 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑁 ∈ (ℤ‘(𝑁𝑗)))
16 fzss2 12495 . . . 4 (𝑁 ∈ (ℤ‘(𝑁𝑗)) → (0...(𝑁𝑗)) ⊆ (0...𝑁))
1715, 16syl 17 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (0...(𝑁𝑗)) ⊆ (0...𝑁))
18 simpr 479 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ (0...(𝑁𝑗)))
1917, 18sseldd 3710 . 2 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ (0...𝑁))
20 elfzelz 12456 . . . . . 6 (𝑘 ∈ (0...(𝑁𝑗)) → 𝑘 ∈ ℤ)
2120adantl 473 . . . . 5 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ ℤ)
2221zred 11595 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ∈ ℝ)
23 elfzle2 12459 . . . . 5 (𝑘 ∈ (0...(𝑁𝑗)) → 𝑘 ≤ (𝑁𝑗))
2423adantl 473 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑘 ≤ (𝑁𝑗))
2522, 6, 9, 24lesubd 10744 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ≤ (𝑁𝑘))
26 elfzuz 12452 . . . . 5 (𝑗 ∈ (0...𝑁) → 𝑗 ∈ (ℤ‘0))
2726adantr 472 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ (ℤ‘0))
285, 21zsubcld 11600 . . . 4 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑁𝑘) ∈ ℤ)
29 elfz5 12448 . . . 4 ((𝑗 ∈ (ℤ‘0) ∧ (𝑁𝑘) ∈ ℤ) → (𝑗 ∈ (0...(𝑁𝑘)) ↔ 𝑗 ≤ (𝑁𝑘)))
3027, 28, 29syl2anc 696 . . 3 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑗 ∈ (0...(𝑁𝑘)) ↔ 𝑗 ≤ (𝑁𝑘)))
3125, 30mpbird 247 . 2 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → 𝑗 ∈ (0...(𝑁𝑘)))
3219, 31jca 555 1 ((𝑗 ∈ (0...𝑁) ∧ 𝑘 ∈ (0...(𝑁𝑗))) → (𝑘 ∈ (0...𝑁) ∧ 𝑗 ∈ (0...(𝑁𝑘))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2103  wss 3680   class class class wbr 4760  cfv 6001  (class class class)co 6765  0cc0 10049  cle 10188  cmin 10379  0cn0 11405  cz 11490  cuz 11800  ...cfz 12440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441
This theorem is referenced by:  fsum0diag  14629  fprod0diag  14837
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