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Theorem prodmolem3 15969
Description: Lemma for prodmo 15972. (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   𝜑,𝑗   𝑗,𝐺   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 𝑚 𝑖 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mulcl 11239 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
21adantl 481 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3 mulcom 11241 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
43adantl 481 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5 mulass 11243 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
65adantl 481 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
7 prodmolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 494 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 12921 . . . 4 ℕ = (ℤ‘1)
108, 9eleqtrdi 2851 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssidd 4007 . . 3 (𝜑 → ℂ ⊆ ℂ)
12 prodmolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
13 f1ocnv 6860 . . . . . 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 6871 . . . . 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 7464 . . . . . . . . . 10 (1...𝑁) ∈ V
1918f1oen 9013 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2017, 19syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
21 fzfi 14013 . . . . . . . . 9 (1...𝑁) ∈ Fin
22 fzfi 14013 . . . . . . . . 9 (1...𝑀) ∈ Fin
23 hashen 14386 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2421, 22, 23mp2an 692 . . . . . . . 8 ((♯‘(1...𝑁)) = (♯‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2520, 24sylibr 234 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = (♯‘(1...𝑀)))
267simprd 495 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
2726nnnn0d 12587 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
28 hashfz1 14385 . . . . . . . 8 (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
2927, 28syl 17 . . . . . . 7 (𝜑 → (♯‘(1...𝑁)) = 𝑁)
308nnnn0d 12587 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
31 hashfz1 14385 . . . . . . . 8 (𝑀 ∈ ℕ0 → (♯‘(1...𝑀)) = 𝑀)
3230, 31syl 17 . . . . . . 7 (𝜑 → (♯‘(1...𝑀)) = 𝑀)
3325, 29, 323eqtr3rd 2786 . . . . . 6 (𝜑𝑀 = 𝑁)
3433oveq2d 7447 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
3534f1oeq2d 6844 . . . 4 (𝜑 → ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
3617, 35mpbird 257 . . 3 (𝜑 → (𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
37 prodmo.3 . . . . 5 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
38 fveq2 6906 . . . . . 6 (𝑗 = 𝑚 → (𝑓𝑗) = (𝑓𝑚))
3938csbeq1d 3903 . . . . 5 (𝑗 = 𝑚(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
40 elfznn 13593 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4140adantl 481 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
42 f1of 6848 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4312, 42syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4443ffvelcdmda 7104 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
45 prodmo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4645ralrimiva 3146 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4746adantr 480 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
48 nfcsb1v 3923 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4948nfel1 2922 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
50 csbeq1a 3913 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5150eleq1d 2826 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5249, 51rspc 3610 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5344, 47, 52sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
5437, 39, 41, 53fvmptd3 7039 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5554, 53eqeltrd 2841 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
5634f1oeq2d 6844 . . . . . . . . . . 11 (𝜑 → (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑁)–1-1-onto𝐴))
5715, 56mpbird 257 . . . . . . . . . 10 (𝜑𝐾:(1...𝑀)–1-1-onto𝐴)
58 f1of 6848 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
5957, 58syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
60 fvco3 7008 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6159, 60sylan 580 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6261fveq2d 6910 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6312adantr 480 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6459ffvelcdmda 7104 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
65 f1ocnvfv2 7297 . . . . . . . 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, 66eqtrd 2777 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝐾𝑖))
6867csbeq1d 3903 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
6968fveq2d 6910 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
70 f1of 6848 . . . . . . 7 ((𝑓𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7136, 70syl 17 . . . . . 6 (𝜑 → (𝑓𝐾):(1...𝑀)⟶(1...𝑀))
7271ffvelcdmda 7104 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
73 elfznn 13593 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
74 fveq2 6906 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) = (𝑓‘((𝑓𝐾)‘𝑖)))
7574csbeq1d 3903 . . . . . 6 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
7675, 37fvmpti 7015 . . . . 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 13593 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
7978adantl 481 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
80 fveq2 6906 . . . . . . 7 (𝑗 = 𝑖 → (𝐾𝑗) = (𝐾𝑖))
8180csbeq1d 3903 . . . . . 6 (𝑗 = 𝑖(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
82 prodmolem3.4 . . . . . 6 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
8381, 82fvmpti 7015 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8479, 83syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8569, 77, 843eqtr4rd 2788 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
862, 4, 6, 10, 11, 36, 55, 85seqf1o 14084 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
8733fveq2d 6910 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
8886, 87eqtr3d 2779 1 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  csb 3899  ifcif 4525   class class class wbr 5143  cmpt 5225   I cid 5577  ccnv 5684  ccom 5689  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cen 8982  Fincfn 8985  cc 11153  1c1 11156   · cmul 11160  cn 12266  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  seqcseq 14042  chash 14369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370
This theorem is referenced by:  prodmolem2a  15970  prodmo  15972
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