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Theorem fzennn 13789
Description: The cardinality of a finite set of sequential integers. (See om2uz0i 13768 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 7345 . . 3 (𝑛 = 0 → (1...𝑛) = (1...0))
2 fveq2 6825 . . 3 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
31, 2breq12d 5105 . 2 (𝑛 = 0 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...0) ≈ (𝐺‘0)))
4 oveq2 7345 . . 3 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5 fveq2 6825 . . 3 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
64, 5breq12d 5105 . 2 (𝑛 = 𝑚 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑚) ≈ (𝐺𝑚)))
7 oveq2 7345 . . 3 (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8 fveq2 6825 . . 3 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
97, 8breq12d 5105 . 2 (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
10 oveq2 7345 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11 fveq2 6825 . . 3 (𝑛 = 𝑁 → (𝐺𝑛) = (𝐺𝑁))
1210, 11breq12d 5105 . 2 (𝑛 = 𝑁 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑁) ≈ (𝐺𝑁)))
13 0ex 5251 . . . 4 ∅ ∈ V
1413enref 8846 . . 3 ∅ ≈ ∅
15 fz10 13378 . . 3 (1...0) = ∅
16 0z 12431 . . . . . 6 0 ∈ ℤ
17 fzennn.1 . . . . . 6 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 0) ↾ ω)
1816, 17om2uzf1oi 13774 . . . . 5 𝐺:ω–1-1-onto→(ℤ‘0)
19 peano1 7803 . . . . 5 ∅ ∈ ω
2018, 19pm3.2i 471 . . . 4 (𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω)
2116, 17om2uz0i 13768 . . . 4 (𝐺‘∅) = 0
22 f1ocnvfv 7206 . . . 4 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (𝐺‘0) = ∅))
2320, 21, 22mp2 9 . . 3 (𝐺‘0) = ∅
2414, 15, 233brtr4i 5122 . 2 (1...0) ≈ (𝐺‘0)
25 simpr 485 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...𝑚) ≈ (𝐺𝑚))
26 ovex 7370 . . . . . . 7 (𝑚 + 1) ∈ V
27 fvex 6838 . . . . . . 7 (𝐺𝑚) ∈ V
28 en2sn 8906 . . . . . . 7 (((𝑚 + 1) ∈ V ∧ (𝐺𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
2926, 27, 28mp2an 689 . . . . . 6 {(𝑚 + 1)} ≈ {(𝐺𝑚)}
3029a1i 11 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
31 fzp1disj 13416 . . . . . 6 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
3231a1i 11 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
33 f1ocnvdm 7213 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺𝑚) ∈ ω)
3418, 33mpan 687 . . . . . . . . 9 (𝑚 ∈ (ℤ‘0) → (𝐺𝑚) ∈ ω)
35 nn0uz 12721 . . . . . . . . 9 0 = (ℤ‘0)
3634, 35eleq2s 2855 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺𝑚) ∈ ω)
37 nnord 7788 . . . . . . . 8 ((𝐺𝑚) ∈ ω → Ord (𝐺𝑚))
38 ordirr 6320 . . . . . . . 8 (Ord (𝐺𝑚) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
3936, 37, 383syl 18 . . . . . . 7 (𝑚 ∈ ℕ0 → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
4039adantr 481 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
41 disjsn 4659 . . . . . 6 (((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅ ↔ ¬ (𝐺𝑚) ∈ (𝐺𝑚))
4240, 41sylibr 233 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)
43 unen 8911 . . . . 5 ((((1...𝑚) ≈ (𝐺𝑚) ∧ {(𝑚 + 1)} ≈ {(𝐺𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
4425, 30, 32, 42, 43syl22anc 836 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
45 1z 12451 . . . . . 6 1 ∈ ℤ
46 1m1e0 12146 . . . . . . . . . 10 (1 − 1) = 0
4746fveq2i 6828 . . . . . . . . 9 (ℤ‘(1 − 1)) = (ℤ‘0)
4835, 47eqtr4i 2767 . . . . . . . 8 0 = (ℤ‘(1 − 1))
4948eleq2i 2828 . . . . . . 7 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
5049biimpi 215 . . . . . 6 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
51 fzsuc2 13415 . . . . . 6 ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
5245, 50, 51sylancr 587 . . . . 5 (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
5352adantr 481 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
54 peano2 7805 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → suc (𝐺𝑚) ∈ ω)
5536, 54syl 17 . . . . . . . 8 (𝑚 ∈ ℕ0 → suc (𝐺𝑚) ∈ ω)
5655, 18jctil 520 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω))
5716, 17om2uzsuci 13769 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
5836, 57syl 17 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
5935eleq2i 2828 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
6059biimpi 215 . . . . . . . . . 10 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
61 f1ocnvfv2 7205 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺‘(𝐺𝑚)) = 𝑚)
6218, 60, 61sylancr 587 . . . . . . . . 9 (𝑚 ∈ ℕ0 → (𝐺‘(𝐺𝑚)) = 𝑚)
6362oveq1d 7352 . . . . . . . 8 (𝑚 ∈ ℕ0 → ((𝐺‘(𝐺𝑚)) + 1) = (𝑚 + 1))
6458, 63eqtrd 2776 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = (𝑚 + 1))
65 f1ocnvfv 7206 . . . . . . 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 481 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
68 df-suc 6308 . . . . 5 suc (𝐺𝑚) = ((𝐺𝑚) ∪ {(𝐺𝑚)})
6967, 68eqtrdi 2792 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = ((𝐺𝑚) ∪ {(𝐺𝑚)}))
7044, 53, 693brtr4d 5124 . . 3 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1)))
7170ex 413 . 2 (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (𝐺𝑚) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
723, 6, 9, 12, 24, 71nn0ind 12516 1 (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1540  wcel 2105  Vcvv 3441  cun 3896  cin 3897  c0 4269  {csn 4573   class class class wbr 5092  cmpt 5175  ccnv 5619  cres 5622  Ord word 6301  suc csuc 6304  1-1-ontowf1o 6478  cfv 6479  (class class class)co 7337  ωcom 7780  reccrdg 8310  cen 8801  0cc0 10972  1c1 10973   + caddc 10975  cmin 11306  0cn0 12334  cz 12420  cuz 12683  ...cfz 13340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-er 8569  df-en 8805  df-dom 8806  df-sdom 8807  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-n0 12335  df-z 12421  df-uz 12684  df-fz 13341
This theorem is referenced by:  fzen2  13790  cardfz  13791
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