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Theorem prodmolem3 15279
Description: Lemma for prodmo 15282. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
prodmo.1 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
prodmo.2 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
prodmo.3 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
prodmolem3.4 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
prodmolem3.5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
prodmolem3.6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
prodmolem3.7 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
Assertion
Ref Expression
prodmolem3 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝜑,𝑘   𝐵,𝑗   𝑓,𝑗,𝑘   𝑗,𝐺   𝑗,𝑘,𝜑   𝑗,𝐾   𝑗,𝑀
Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓,𝑗)   𝐵(𝑓,𝑘)   𝐹(𝑓,𝑗)   𝐺(𝑓,𝑘)   𝐻(𝑓,𝑗,𝑘)   𝐾(𝑓,𝑘)   𝑀(𝑓,𝑘)   𝑁(𝑓,𝑗,𝑘)

Proof of Theorem prodmolem3
Dummy variables 𝑖 𝑚 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulcl 10610 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
21adantl 485 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3 mulcom 10612 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
43adantl 485 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5 mulass 10614 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
65adantl 485 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
7 prodmolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 498 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 12269 . . . 4 ℕ = (ℤ‘1)
108, 9eleqtrdi 2900 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssidd 3938 . . 3 (𝜑 → ℂ ⊆ ℂ)
12 prodmolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
13 f1ocnv 6602 . . . . . 6 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:𝐴1-1-onto→(1...𝑀))
1412, 13syl 17 . . . . 5 (𝜑𝑓:𝐴1-1-onto→(1...𝑀))
15 prodmolem3.7 . . . . 5 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
16 f1oco 6612 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1714, 15, 16syl2anc 587 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
18 ovex 7168 . . . . . . . . . 10 (1...𝑁) ∈ V
1918f1oen 8513 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2017, 19syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
21 fzfi 13335 . . . . . . . . 9 (1...𝑁) ∈ Fin
22 fzfi 13335 . . . . . . . . 9 (1...𝑀) ∈ Fin
23 hashen 13703 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2421, 22, 23mp2an 691 . . . . . . . 8 ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2520, 24sylibr 237 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
267simprd 499 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
2726nnnn0d 11943 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
28 hashfz1 13702 . . . . . . . 8 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2927, 28syl 17 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
308nnnn0d 11943 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
31 hashfz1 13702 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
3230, 31syl 17 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
3325, 29, 323eqtr3rd 2842 . . . . . 6 (𝜑𝑀 = 𝑁)
3433oveq2d 7151 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
3534f1oeq2d 6586 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3617, 35mpbird 260 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37 prodmo.3 . . . . 5 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
38 fveq2 6645 . . . . . 6 (𝑗 = 𝑚 → (𝑓𝑗) = (𝑓𝑚))
3938csbeq1d 3832 . . . . 5 (𝑗 = 𝑚(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
40 elfznn 12931 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4140adantl 485 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42 f1of 6590 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4312, 42syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4443ffvelrnda 6828 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
45 prodmo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4645ralrimiva 3149 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4746adantr 484 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
48 nfcsb1v 3852 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4948nfel1 2971 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
50 csbeq1a 3842 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5150eleq1d 2874 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5249, 51rspc 3559 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5344, 47, 52sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
5437, 39, 41, 53fvmptd3 6768 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5554, 53eqeltrd 2890 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
5634f1oeq2d 6586 . . . . . . . . . . 11 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
5715, 56mpbird 260 . . . . . . . . . 10 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
58 f1of 6590 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
5957, 58syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
60 fvco3 6737 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6159, 60sylan 583 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6261fveq2d 6649 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6312adantr 484 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6459ffvelrnda 6828 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
65 f1ocnvfv2 7012 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
6663, 64, 65syl2anc 587 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
6762, 66eqtrd 2833 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝐾𝑖))
6867csbeq1d 3832 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
6968fveq2d 6649 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
70 f1of 6590 . . . . . . 7 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7136, 70syl 17 . . . . . 6 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7271ffvelrnda 6828 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
73 elfznn 12931 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
74 fveq2 6645 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) = (𝑓‘((𝑓𝐾)‘𝑖)))
7574csbeq1d 3832 . . . . . 6 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
7675, 37fvmpti 6744 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ ℕ → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
7772, 73, 763syl 18 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
78 elfznn 12931 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
7978adantl 485 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
80 fveq2 6645 . . . . . . 7 (𝑗 = 𝑖 → (𝐾𝑗) = (𝐾𝑖))
8180csbeq1d 3832 . . . . . 6 (𝑗 = 𝑖(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
82 prodmolem3.4 . . . . . 6 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
8381, 82fvmpti 6744 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8479, 83syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8569, 77, 843eqtr4rd 2844 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
862, 4, 6, 10, 11, 36, 55, 85seqf1o 13407 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
8733fveq2d 6649 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
8886, 87eqtr3d 2835 1 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  csb 3828  ifcif 4425   class class class wbr 5030  cmpt 5110   I cid 5424  ccnv 5518  ccom 5523  wf 6320  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135  cen 8489  Fincfn 8492  cc 10524  1c1 10527   · cmul 10531  cn 11625  0cn0 11885  cz 11969  cuz 12231  ...cfz 12885  seqcseq 13364  chash 13686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-seq 13365  df-hash 13687
This theorem is referenced by:  prodmolem2a  15280  prodmo  15282
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