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

Theorem fzennn 13933
Description: The cardinality of a finite set of sequential integers. (See om2uz0i 13912 for a description of the hypothesis.) (Contributed by Mario Carneiro, 12-Feb-2013.) (Revised by Mario Carneiro, 7-Mar-2014.)
Hypothesis
Ref Expression
fzennn.1 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
Assertion
Ref Expression
fzennn (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))

Proof of Theorem fzennn
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7417 . . 3 (𝑛 = 0 → (1...𝑛) = (1...0))
2 fveq2 6892 . . 3 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
31, 2breq12d 5162 . 2 (𝑛 = 0 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...0) ≈ (𝐺‘0)))
4 oveq2 7417 . . 3 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5 fveq2 6892 . . 3 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
64, 5breq12d 5162 . 2 (𝑛 = 𝑚 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑚) ≈ (𝐺𝑚)))
7 oveq2 7417 . . 3 (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8 fveq2 6892 . . 3 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
97, 8breq12d 5162 . 2 (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
10 oveq2 7417 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11 fveq2 6892 . . 3 (𝑛 = 𝑁 → (𝐺𝑛) = (𝐺𝑁))
1210, 11breq12d 5162 . 2 (𝑛 = 𝑁 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑁) ≈ (𝐺𝑁)))
13 0ex 5308 . . . 4 ∅ ∈ V
1413enref 8981 . . 3 ∅ ≈ ∅
15 fz10 13522 . . 3 (1...0) = ∅
16 0z 12569 . . . . . 6 0 ∈ ℤ
17 fzennn.1 . . . . . 6 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
1816, 17om2uzf1oi 13918 . . . . 5 𝐺:ω–1-1-onto→(ℤ‘0)
19 peano1 7879 . . . . 5 ∅ ∈ ω
2018, 19pm3.2i 472 . . . 4 (𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω)
2116, 17om2uz0i 13912 . . . 4 (𝐺‘∅) = 0
22 f1ocnvfv 7276 . . . 4 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (𝐺‘0) = ∅))
2320, 21, 22mp2 9 . . 3 (𝐺‘0) = ∅
2414, 15, 233brtr4i 5179 . 2 (1...0) ≈ (𝐺‘0)
25 simpr 486 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...𝑚) ≈ (𝐺𝑚))
26 ovex 7442 . . . . . . 7 (𝑚 + 1) ∈ V
27 fvex 6905 . . . . . . 7 (𝐺𝑚) ∈ V
28 en2sn 9041 . . . . . . 7 (((𝑚 + 1) ∈ V ∧ (𝐺𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
2926, 27, 28mp2an 691 . . . . . 6 {(𝑚 + 1)} ≈ {(𝐺𝑚)}
3029a1i 11 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
31 fzp1disj 13560 . . . . . 6 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
3231a1i 11 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
33 f1ocnvdm 7283 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺𝑚) ∈ ω)
3418, 33mpan 689 . . . . . . . . 9 (𝑚 ∈ (ℤ‘0) → (𝐺𝑚) ∈ ω)
35 nn0uz 12864 . . . . . . . . 9 0 = (ℤ‘0)
3634, 35eleq2s 2852 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺𝑚) ∈ ω)
37 nnord 7863 . . . . . . . 8 ((𝐺𝑚) ∈ ω → Ord (𝐺𝑚))
38 ordirr 6383 . . . . . . . 8 (Ord (𝐺𝑚) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
3936, 37, 383syl 18 . . . . . . 7 (𝑚 ∈ ℕ0 → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
4039adantr 482 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
41 disjsn 4716 . . . . . 6 (((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅ ↔ ¬ (𝐺𝑚) ∈ (𝐺𝑚))
4240, 41sylibr 233 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)
43 unen 9046 . . . . 5 ((((1...𝑚) ≈ (𝐺𝑚) ∧ {(𝑚 + 1)} ≈ {(𝐺𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
4425, 30, 32, 42, 43syl22anc 838 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
45 1z 12592 . . . . . 6 1 ∈ ℤ
46 1m1e0 12284 . . . . . . . . . 10 (1 − 1) = 0
4746fveq2i 6895 . . . . . . . . 9 (ℤ‘(1 − 1)) = (ℤ‘0)
4835, 47eqtr4i 2764 . . . . . . . 8 0 = (ℤ‘(1 − 1))
4948eleq2i 2826 . . . . . . 7 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
5049biimpi 215 . . . . . 6 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
51 fzsuc2 13559 . . . . . 6 ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
5245, 50, 51sylancr 588 . . . . 5 (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
5352adantr 482 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
54 peano2 7881 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → suc (𝐺𝑚) ∈ ω)
5536, 54syl 17 . . . . . . . 8 (𝑚 ∈ ℕ0 → suc (𝐺𝑚) ∈ ω)
5655, 18jctil 521 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω))
5716, 17om2uzsuci 13913 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
5836, 57syl 17 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
5935eleq2i 2826 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
6059biimpi 215 . . . . . . . . . 10 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
61 f1ocnvfv2 7275 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺‘(𝐺𝑚)) = 𝑚)
6218, 60, 61sylancr 588 . . . . . . . . 9 (𝑚 ∈ ℕ0 → (𝐺‘(𝐺𝑚)) = 𝑚)
6362oveq1d 7424 . . . . . . . 8 (𝑚 ∈ ℕ0 → ((𝐺‘(𝐺𝑚)) + 1) = (𝑚 + 1))
6458, 63eqtrd 2773 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = (𝑚 + 1))
65 f1ocnvfv 7276 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω) → ((𝐺‘suc (𝐺𝑚)) = (𝑚 + 1) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚)))
6656, 64, 65sylc 65 . . . . . 6 (𝑚 ∈ ℕ0 → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
6766adantr 482 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
68 df-suc 6371 . . . . 5 suc (𝐺𝑚) = ((𝐺𝑚) ∪ {(𝐺𝑚)})
6967, 68eqtrdi 2789 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = ((𝐺𝑚) ∪ {(𝐺𝑚)}))
7044, 53, 693brtr4d 5181 . . 3 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1)))
7170ex 414 . 2 (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (𝐺𝑚) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
723, 6, 9, 12, 24, 71nn0ind 12657 1 (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cun 3947  cin 3948  c0 4323  {csn 4629   class class class wbr 5149  cmpt 5232  ccnv 5676  cres 5679  Ord word 6364  suc csuc 6367  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  ωcom 7855  reccrdg 8409  cen 8936  0cc0 11110  1c1 11111   + caddc 11113  cmin 11444  0cn0 12472  cz 12558  cuz 12822  ...cfz 13484
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485
This theorem is referenced by:  fzen2  13934  cardfz  13935
  Copyright terms: Public domain W3C validator