MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  prodmolem3 Structured version   Visualization version   GIF version

Theorem prodmolem3 14599
Description: Lemma for prodmo 14602. (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 9972 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) ∈ ℂ)
21adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) ∈ ℂ)
3 mulcom 9974 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
43adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 · 𝑗) = (𝑗 · 𝑚))
5 mulass 9976 . . . 4 ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
65adantl 482 . . 3 ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑚 · 𝑗) · 𝑧) = (𝑚 · (𝑗 · 𝑧)))
7 prodmolem3.5 . . . . 5 (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
87simpld 475 . . . 4 (𝜑𝑀 ∈ ℕ)
9 nnuz 11675 . . . 4 ℕ = (ℤ‘1)
108, 9syl6eleq 2708 . . 3 (𝜑𝑀 ∈ (ℤ‘1))
11 ssid 3608 . . . 4 ℂ ⊆ ℂ
1211a1i 11 . . 3 (𝜑 → ℂ ⊆ ℂ)
13 prodmolem3.6 . . . . . 6 (𝜑𝑓:(1...𝑀)–1-1-onto𝐴)
14 f1ocnv 6111 . . . . . 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 6121 . . . . 5 ((𝑓:𝐴1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto𝐴) → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
1815, 16, 17syl2anc 692 . . . 4 (𝜑 → (𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
19 ovex 6638 . . . . . . . . . 10 (1...𝑁) ∈ V
2019f1oen 7928 . . . . . . . . 9 ((𝑓𝐾):(1...𝑁)–1-1-onto→(1...𝑀) → (1...𝑁) ≈ (1...𝑀))
2118, 20syl 17 . . . . . . . 8 (𝜑 → (1...𝑁) ≈ (1...𝑀))
22 fzfi 12719 . . . . . . . . 9 (1...𝑁) ∈ Fin
23 fzfi 12719 . . . . . . . . 9 (1...𝑀) ∈ Fin
24 hashen 13083 . . . . . . . . 9 (((1...𝑁) ∈ Fin ∧ (1...𝑀) ∈ Fin) → ((#‘(1...𝑁)) = (#‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀)))
2522, 23, 24mp2an 707 . . . . . . . 8 ((#‘(1...𝑁)) = (#‘(1...𝑀)) ↔ (1...𝑁) ≈ (1...𝑀))
2621, 25sylibr 224 . . . . . . 7 (𝜑 → (#‘(1...𝑁)) = (#‘(1...𝑀)))
277simprd 479 . . . . . . . . 9 (𝜑𝑁 ∈ ℕ)
2827nnnn0d 11303 . . . . . . . 8 (𝜑𝑁 ∈ ℕ0)
29 hashfz1 13082 . . . . . . . 8 (𝑁 ∈ ℕ0 → (#‘(1...𝑁)) = 𝑁)
3028, 29syl 17 . . . . . . 7 (𝜑 → (#‘(1...𝑁)) = 𝑁)
318nnnn0d 11303 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
32 hashfz1 13082 . . . . . . . 8 (𝑀 ∈ ℕ0 → (#‘(1...𝑀)) = 𝑀)
3331, 32syl 17 . . . . . . 7 (𝜑 → (#‘(1...𝑀)) = 𝑀)
3426, 30, 333eqtr3rd 2664 . . . . . 6 (𝜑𝑀 = 𝑁)
3534oveq2d 6626 . . . . 5 (𝜑 → (1...𝑀) = (1...𝑁))
36 f1oeq2 6090 . . . . 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 12320 . . . . . 6 (𝑚 ∈ (1...𝑀) → 𝑚 ∈ ℕ)
4039adantl 482 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
41 f1of 6099 . . . . . . . 8 (𝑓:(1...𝑀)–1-1-onto𝐴𝑓:(1...𝑀)⟶𝐴)
4213, 41syl 17 . . . . . . 7 (𝜑𝑓:(1...𝑀)⟶𝐴)
4342ffvelrnda 6320 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) ∈ 𝐴)
44 prodmo.2 . . . . . . . 8 ((𝜑𝑘𝐴) → 𝐵 ∈ ℂ)
4544ralrimiva 2961 . . . . . . 7 (𝜑 → ∀𝑘𝐴 𝐵 ∈ ℂ)
4645adantr 481 . . . . . 6 ((𝜑𝑚 ∈ (1...𝑀)) → ∀𝑘𝐴 𝐵 ∈ ℂ)
47 nfcsb1v 3534 . . . . . . . 8 𝑘(𝑓𝑚) / 𝑘𝐵
4847nfel1 2775 . . . . . . 7 𝑘(𝑓𝑚) / 𝑘𝐵 ∈ ℂ
49 csbeq1a 3527 . . . . . . . 8 (𝑘 = (𝑓𝑚) → 𝐵 = (𝑓𝑚) / 𝑘𝐵)
5049eleq1d 2683 . . . . . . 7 (𝑘 = (𝑓𝑚) → (𝐵 ∈ ℂ ↔ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5148, 50rspc 3292 . . . . . 6 ((𝑓𝑚) ∈ 𝐴 → (∀𝑘𝐴 𝐵 ∈ ℂ → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ))
5243, 46, 51sylc 65 . . . . 5 ((𝜑𝑚 ∈ (1...𝑀)) → (𝑓𝑚) / 𝑘𝐵 ∈ ℂ)
53 fveq2 6153 . . . . . . 7 (𝑗 = 𝑚 → (𝑓𝑗) = (𝑓𝑚))
5453csbeq1d 3525 . . . . . 6 (𝑗 = 𝑚(𝑓𝑗) / 𝑘𝐵 = (𝑓𝑚) / 𝑘𝐵)
55 prodmo.3 . . . . . 6 𝐺 = (𝑗 ∈ ℕ ↦ (𝑓𝑗) / 𝑘𝐵)
5654, 55fvmptg 6242 . . . . 5 ((𝑚 ∈ ℕ ∧ (𝑓𝑚) / 𝑘𝐵 ∈ ℂ) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5740, 52, 56syl2anc 692 . . . 4 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) = (𝑓𝑚) / 𝑘𝐵)
5857, 52eqeltrd 2698 . . 3 ((𝜑𝑚 ∈ (1...𝑀)) → (𝐺𝑚) ∈ ℂ)
59 f1oeq2 6090 . . . . . . . . . . . 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 6099 . . . . . . . . . 10 (𝐾:(1...𝑀)–1-1-onto𝐴𝐾:(1...𝑀)⟶𝐴)
6361, 62syl 17 . . . . . . . . 9 (𝜑𝐾:(1...𝑀)⟶𝐴)
64 fvco3 6237 . . . . . . . . 9 ((𝐾:(1...𝑀)⟶𝐴𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6563, 64sylan 488 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) = (𝑓‘(𝐾𝑖)))
6665fveq2d 6157 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝑓‘(𝑓‘(𝐾𝑖))))
6713adantr 481 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto𝐴)
6863ffvelrnda 6320 . . . . . . . 8 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐾𝑖) ∈ 𝐴)
69 f1ocnvfv2 6493 . . . . . . . 8 ((𝑓:(1...𝑀)–1-1-onto𝐴 ∧ (𝐾𝑖) ∈ 𝐴) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7067, 68, 69syl2anc 692 . . . . . . 7 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘(𝑓‘(𝐾𝑖))) = (𝐾𝑖))
7166, 70eqtrd 2655 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) = (𝐾𝑖))
7271csbeq1d 3525 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
7372fveq2d 6157 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → ( I ‘(𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
74 f1of 6099 . . . . . . 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 6320 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ((𝑓𝐾)‘𝑖) ∈ (1...𝑀))
77 elfznn 12320 . . . . 5 (((𝑓𝐾)‘𝑖) ∈ (1...𝑀) → ((𝑓𝐾)‘𝑖) ∈ ℕ)
78 fveq2 6153 . . . . . . 7 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) = (𝑓‘((𝑓𝐾)‘𝑖)))
7978csbeq1d 3525 . . . . . 6 (𝑗 = ((𝑓𝐾)‘𝑖) → (𝑓𝑗) / 𝑘𝐵 = (𝑓‘((𝑓𝐾)‘𝑖)) / 𝑘𝐵)
8079, 55fvmpti 6243 . . . . 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 12320 . . . . . 6 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
8382adantl 482 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → 𝑖 ∈ ℕ)
84 fveq2 6153 . . . . . . 7 (𝑗 = 𝑖 → (𝐾𝑗) = (𝐾𝑖))
8584csbeq1d 3525 . . . . . 6 (𝑗 = 𝑖(𝐾𝑗) / 𝑘𝐵 = (𝐾𝑖) / 𝑘𝐵)
86 prodmolem3.4 . . . . . 6 𝐻 = (𝑗 ∈ ℕ ↦ (𝐾𝑗) / 𝑘𝐵)
8785, 86fvmpti 6243 . . . . 5 (𝑖 ∈ ℕ → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8883, 87syl 17 . . . 4 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = ( I ‘(𝐾𝑖) / 𝑘𝐵))
8973, 81, 883eqtr4rd 2666 . . 3 ((𝜑𝑖 ∈ (1...𝑀)) → (𝐻𝑖) = (𝐺‘((𝑓𝐾)‘𝑖)))
902, 4, 6, 10, 12, 38, 58, 89seqf1o 12790 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐺)‘𝑀))
9134fveq2d 6157 . 2 (𝜑 → (seq1( · , 𝐻)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
9290, 91eqtr3d 2657 1 (𝜑 → (seq1( · , 𝐺)‘𝑀) = (seq1( · , 𝐻)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  csb 3518  wss 3559  ifcif 4063   class class class wbr 4618  cmpt 4678   I cid 4989  ccnv 5078  ccom 5083  wf 5848  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  cen 7904  Fincfn 7907  cc 9886  1c1 9889   · cmul 9893  cn 10972  0cn0 11244  cz 11329  cuz 11639  ...cfz 12276  seqcseq 12749  #chash 13065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066
This theorem is referenced by:  prodmolem2a  14600  prodmo  14602
  Copyright terms: Public domain W3C validator