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Theorem prodmolem3 14870
Description: Lemma for prodmo 14873. (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 10222 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
21adantl 467 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3 mulcom 10224 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
43adantl 467 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5 mulass 10226 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
65adantl 467 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
7 prodmolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 482 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 11925 . . . 4 ℕ = (ℤ‘1)
108, 9syl6eleq 2860 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssid 3773 . . . 4 ℂ ⊆ ℂ
1211a1i 11 . . 3 (𝜑 → ℂ ⊆ ℂ)
13 prodmolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
14 f1ocnv 6290 . . . . . 6 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:𝐴1-1-onto→(1...𝑀))
1513, 14syl 17 . . . . 5 (𝜑𝑓:𝐴1-1-onto→(1...𝑀))
16 prodmolem3.7 . . . . 5 (𝜑𝐾:(1...𝑁)–1-1-onto𝐴)
17 f1oco 6300 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1815, 16, 17syl2anc 573 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
19 ovex 6823 . . . . . . . . . 10 (1...𝑁) ∈ V
2019f1oen 8130 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2118, 20syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
22 fzfi 12979 . . . . . . . . 9 (1...𝑁) ∈ Fin
23 fzfi 12979 . . . . . . . . 9 (1...𝑀) ∈ Fin
24 hashen 13339 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2522, 23, 24mp2an 672 . . . . . . . 8 ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2621, 25sylibr 224 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
277simprd 483 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
2827nnnn0d 11553 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
29 hashfz1 13338 . . . . . . . 8 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
3028, 29syl 17 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
318nnnn0d 11553 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
32 hashfz1 13338 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
3331, 32syl 17 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
3426, 30, 333eqtr3rd 2814 . . . . . 6 (𝜑𝑀 = 𝑁)
3534oveq2d 6809 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
36 f1oeq2 6269 . . . . 5 ((1...𝑀) = (1...𝑁) → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3735, 36syl 17 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3818, 37mpbird 247 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
39 elfznn 12577 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4039adantl 467 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
41 f1of 6278 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4213, 41syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4342ffvelrnda 6502 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
44 prodmo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4544ralrimiva 3115 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4645adantr 466 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
47 nfcsb1v 3698 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4847nfel1 2928 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
49 csbeq1a 3691 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5049eleq1d 2835 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5148, 50rspc 3454 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5243, 46, 51sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
53 fveq2 6332 . . . . . . 7 (𝑗 = 𝑚 → (𝑓𝑗) = (𝑓𝑚))
5453csbeq1d 3689 . . . . . 6 (𝑗 = 𝑚(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
55 prodmo.3 . . . . . 6 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
5654, 55fvmptg 6422 . . . . 5 ((𝑚 ∈ ℕ ∧ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5740, 52, 56syl2anc 573 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5857, 52eqeltrd 2850 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
59 f1oeq2 6269 . . . . . . . . . . . 12 ((1...𝑀) = (1...𝑁) → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
6035, 59syl 17 . . . . . . . . . . 11 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
6116, 60mpbird 247 . . . . . . . . . 10 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
62 f1of 6278 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
64 fvco3 6417 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6563, 64sylan 569 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6665fveq2d 6336 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6713adantr 466 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6863ffvelrnda 6502 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
69 f1ocnvfv2 6676 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7067, 68, 69syl2anc 573 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7166, 70eqtrd 2805 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝐾𝑖))
7271csbeq1d 3689 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
7372fveq2d 6336 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
74 f1of 6278 . . . . . . 7 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7538, 74syl 17 . . . . . 6 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7675ffvelrnda 6502 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
77 elfznn 12577 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
78 fveq2 6332 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) = (𝑓‘((𝑓𝐾)‘𝑖)))
7978csbeq1d 3689 . . . . . 6 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
8079, 55fvmpti 6423 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ ℕ → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
8176, 77, 803syl 18 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐺‘((𝑓𝐾)‘𝑖)) = ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵))
82 elfznn 12577 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
8382adantl 467 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
84 fveq2 6332 . . . . . . 7 (𝑗 = 𝑖 → (𝐾𝑗) = (𝐾𝑖))
8584csbeq1d 3689 . . . . . 6 (𝑗 = 𝑖(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
86 prodmolem3.4 . . . . . 6 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
8785, 86fvmpti 6423 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8883, 87syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8973, 81, 883eqtr4rd 2816 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
902, 4, 6, 10, 12, 38, 58, 89seqf1o 13049 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
9134fveq2d 6336 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
9290, 91eqtr3d 2807 1 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  csb 3682  wss 3723  ifcif 4225   class class class wbr 4786  cmpt 4863   I cid 5156  ccnv 5248  ccom 5253  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  cen 8106  Fincfn 8109  cc 10136  1c1 10139   · cmul 10143  cn 11222  0cn0 11494  cz 11579  cuz 11888  ...cfz 12533  seqcseq 13008  chash 13321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322
This theorem is referenced by:  prodmolem2a  14871  prodmo  14873
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