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Theorem summolem3 15426
Description: Lemma for summo 15429. (Contributed by Mario Carneiro, 29-Mar-2014.)
Hypotheses
Ref Expression
summo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 0))
summo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
summo.3 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
summolem3.4 𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)
summolem3.5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
summolem3.6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
summolem3.7 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
Assertion
Ref Expression
summolem3 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Distinct variable groups:   𝑓,𝑘,𝑛,𝐴   𝑓,𝐹,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝐾,𝑛   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝐵,𝑓,𝑛   𝑘,𝑀,𝑛
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑘)   𝐺(𝑓)   𝐻(𝑓,𝑘,𝑛)   𝐾(𝑓)   𝑀(𝑓)   𝑁(𝑓)

Proof of Theorem summolem3
Dummy variables 𝑖 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 10953 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
21adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3 addcom 11161 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
43adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5 addass 10958 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
65adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7 summolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 495 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 12621 . . . 4 ℕ = (ℤ‘1)
108, 9eleqtrdi 2849 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssidd 3944 . . 3 (𝜑 → ℂ ⊆ ℂ)
12 summolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
13 f1ocnv 6728 . . . . . 6 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:𝐴1-1-onto→(1...𝑀))
1412, 13syl 17 . . . . 5 (𝜑𝑓:𝐴1-1-onto→(1...𝑀))
15 summolem3.7 . . . . 5 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
16 f1oco 6739 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1714, 15, 16syl2anc 584 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18 ovex 7308 . . . . . . . . . 10 (1...𝑁) ∈ V
1918f1oen 8761 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2017, 19syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
21 fzfi 13692 . . . . . . . . 9 (1...𝑁) ∈ Fin
22 fzfi 13692 . . . . . . . . 9 (1...𝑀) ∈ Fin
23 hashen 14061 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2421, 22, 23mp2an 689 . . . . . . . 8 ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2520, 24sylibr 233 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
267simprd 496 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
27 nnnn0 12240 . . . . . . . 8 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
28 hashfz1 14060 . . . . . . . 8 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2926, 27, 283syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
30 nnnn0 12240 . . . . . . . 8 (𝑀 ∈ ℕ → 𝑀 ∈ ℕ0)
31 hashfz1 14060 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
328, 30, 313syl 18 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
3325, 29, 323eqtr3rd 2787 . . . . . 6 (𝜑𝑀 = 𝑁)
3433oveq2d 7291 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
3534f1oeq2d 6712 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3617, 35mpbird 256 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37 summo.3 . . . . 5 𝐺 = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
38 fveq2 6774 . . . . . 6 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
3938csbeq1d 3836 . . . . 5 (𝑛 = 𝑚(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
40 elfznn 13285 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4140adantl 482 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42 f1of 6716 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4312, 42syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4443ffvelrnda 6961 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
45 summo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4645ralrimiva 3103 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4746adantr 481 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
48 nfcsb1v 3857 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4948nfel1 2923 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
50 csbeq1a 3846 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5150eleq1d 2823 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5249, 51rspc 3549 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5344, 47, 52sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
5437, 39, 41, 53fvmptd3 6898 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5554, 53eqeltrd 2839 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
5634f1oeq2d 6712 . . . . . . . . . . 11 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
5715, 56mpbird 256 . . . . . . . . . 10 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
58 f1of 6716 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
5957, 58syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
60 fvco3 6867 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6159, 60sylan 580 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6261fveq2d 6778 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6312adantr 481 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6459ffvelrnda 6961 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
65 f1ocnvfv2 7149 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
6663, 64, 65syl2anc 584 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
6762, 66eqtr2d 2779 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) = (𝑓‘((𝑓𝐾)‘𝑖)))
6867csbeq1d 3836 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
6968fveq2d 6778 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝐾𝑖) / 𝑘𝐵) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
70 elfznn 13285 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
7170adantl 482 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
72 fveq2 6774 . . . . . . 7 (𝑛 = 𝑖 → (𝐾𝑛) = (𝐾𝑖))
7372csbeq1d 3836 . . . . . 6 (𝑛 = 𝑖(𝐾𝑛) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
74 summolem3.4 . . . . . 6 𝐻 = (𝑛 ∈ ℕ ↦ (𝐾𝑛) / 𝑘𝐵)
7573, 74fvmpti 6874 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
7671, 75syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
77 f1of 6716 . . . . . . 7 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7836, 77syl 17 . . . . . 6 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7978ffvelrnda 6961 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
80 elfznn 13285 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
81 fveq2 6774 . . . . . . 7 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) = (𝑓‘((𝑓𝐾)‘𝑖)))
8281csbeq1d 3836 . . . . . 6 (𝑛 = ((𝑓𝐾)‘𝑖) → (𝑓𝑛) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
8382, 37fvmpti 6874 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ ℕ → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8479, 80, 833syl 18 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8569, 76, 843eqtr4d 2788 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
862, 4, 6, 10, 11, 36, 55, 85seqf1o 13764 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
8733fveq2d 6778 . 2 (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
8886, 87eqtr3d 2780 1 (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  csb 3832  ifcif 4459   class class class wbr 5074  cmpt 5157   I cid 5488  ccnv 5588  ccom 5593  wf 6429  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275  cen 8730  Fincfn 8733  cc 10869  0cc0 10871  1c1 10872   + caddc 10874  cn 11973  0cn0 12233  cz 12319  cuz 12582  ...cfz 13239  seqcseq 13721  chash 14044
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-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
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-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045
This theorem is referenced by:  summolem2a  15427  summo  15429
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