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Theorem frecfzennn 10426
Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10399 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)
Hypothesis
Ref Expression
frecfzennn.1 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
Assertion
Ref Expression
frecfzennn (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))

Proof of Theorem frecfzennn
Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5883 . . 3 (𝑛 = 0 → (1...𝑛) = (1...0))
2 fveq2 5516 . . 3 (𝑛 = 0 → (𝐺𝑛) = (𝐺‘0))
31, 2breq12d 4017 . 2 (𝑛 = 0 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...0) ≈ (𝐺‘0)))
4 oveq2 5883 . . 3 (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5 fveq2 5516 . . 3 (𝑛 = 𝑚 → (𝐺𝑛) = (𝐺𝑚))
64, 5breq12d 4017 . 2 (𝑛 = 𝑚 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑚) ≈ (𝐺𝑚)))
7 oveq2 5883 . . 3 (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8 fveq2 5516 . . 3 (𝑛 = (𝑚 + 1) → (𝐺𝑛) = (𝐺‘(𝑚 + 1)))
97, 8breq12d 4017 . 2 (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
10 oveq2 5883 . . 3 (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11 fveq2 5516 . . 3 (𝑛 = 𝑁 → (𝐺𝑛) = (𝐺𝑁))
1210, 11breq12d 4017 . 2 (𝑛 = 𝑁 → ((1...𝑛) ≈ (𝐺𝑛) ↔ (1...𝑁) ≈ (𝐺𝑁)))
13 0ex 4131 . . . 4 ∅ ∈ V
1413enref 6765 . . 3 ∅ ≈ ∅
15 fz10 10046 . . 3 (1...0) = ∅
16 0zd 9265 . . . . . . 7 (⊤ → 0 ∈ ℤ)
17 frecfzennn.1 . . . . . . 7 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
1816, 17frec2uzf1od 10406 . . . . . 6 (⊤ → 𝐺:ω–1-1-onto→(ℤ‘0))
1918mptru 1362 . . . . 5 𝐺:ω–1-1-onto→(ℤ‘0)
20 peano1 4594 . . . . 5 ∅ ∈ ω
2119, 20pm3.2i 272 . . . 4 (𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω)
2216, 17frec2uz0d 10399 . . . . 5 (⊤ → (𝐺‘∅) = 0)
2322mptru 1362 . . . 4 (𝐺‘∅) = 0
24 f1ocnvfv 5780 . . . 4 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (𝐺‘0) = ∅))
2521, 23, 24mp2 16 . . 3 (𝐺‘0) = ∅
2614, 15, 253brtr4i 4034 . 2 (1...0) ≈ (𝐺‘0)
27 simpr 110 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...𝑚) ≈ (𝐺𝑚))
28 peano2nn0 9216 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29 zex 9262 . . . . . . . . . . . . . . 15 ℤ ∈ V
3029mptex 5743 . . . . . . . . . . . . . 14 (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31 vex 2741 . . . . . . . . . . . . . 14 𝑧 ∈ V
3230, 31fvex 5536 . . . . . . . . . . . . 13 ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
3332ax-gen 1449 . . . . . . . . . . . 12 𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34 0z 9264 . . . . . . . . . . . 12 0 ∈ ℤ
35 frecfnom 6402 . . . . . . . . . . . 12 ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3633, 34, 35mp2an 426 . . . . . . . . . . 11 frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
3717fneq1i 5311 . . . . . . . . . . 11 (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
3836, 37mpbir 146 . . . . . . . . . 10 𝐺 Fn ω
39 omex 4593 . . . . . . . . . 10 ω ∈ V
40 fnex 5739 . . . . . . . . . 10 ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
4138, 39, 40mp2an 426 . . . . . . . . 9 𝐺 ∈ V
4241cnvex 5168 . . . . . . . 8 𝐺 ∈ V
43 vex 2741 . . . . . . . 8 𝑚 ∈ V
4442, 43fvex 5536 . . . . . . 7 (𝐺𝑚) ∈ V
45 en2sn 6813 . . . . . . 7 (((𝑚 + 1) ∈ ℕ0 ∧ (𝐺𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4628, 44, 45sylancl 413 . . . . . 6 (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
4746adantr 276 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → {(𝑚 + 1)} ≈ {(𝐺𝑚)})
48 fzp1disj 10080 . . . . . 6 ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
4948a1i 9 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50 f1ocnvdm 5782 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺𝑚) ∈ ω)
5119, 50mpan 424 . . . . . . . . 9 (𝑚 ∈ (ℤ‘0) → (𝐺𝑚) ∈ ω)
52 nn0uz 9562 . . . . . . . . 9 0 = (ℤ‘0)
5351, 52eleq2s 2272 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺𝑚) ∈ ω)
54 nnord 4612 . . . . . . . 8 ((𝐺𝑚) ∈ ω → Ord (𝐺𝑚))
55 ordirr 4542 . . . . . . . 8 (Ord (𝐺𝑚) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5653, 54, 553syl 17 . . . . . . 7 (𝑚 ∈ ℕ0 → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5756adantr 276 . . . . . 6 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ¬ (𝐺𝑚) ∈ (𝐺𝑚))
58 disjsn 3655 . . . . . 6 (((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅ ↔ ¬ (𝐺𝑚) ∈ (𝐺𝑚))
5957, 58sylibr 134 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)
60 unen 6816 . . . . 5 ((((1...𝑚) ≈ (𝐺𝑚) ∧ {(𝑚 + 1)} ≈ {(𝐺𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((𝐺𝑚) ∩ {(𝐺𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
6127, 47, 49, 59, 60syl22anc 1239 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((𝐺𝑚) ∪ {(𝐺𝑚)}))
62 1z 9279 . . . . . 6 1 ∈ ℤ
63 1m1e0 8988 . . . . . . . . . 10 (1 − 1) = 0
6463fveq2i 5519 . . . . . . . . 9 (ℤ‘(1 − 1)) = (ℤ‘0)
6552, 64eqtr4i 2201 . . . . . . . 8 0 = (ℤ‘(1 − 1))
6665eleq2i 2244 . . . . . . 7 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
6766biimpi 120 . . . . . 6 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘(1 − 1)))
68 fzsuc2 10079 . . . . . 6 ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
6962, 67, 68sylancr 414 . . . . 5 (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
7069adantr 276 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
71 peano2 4595 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → suc (𝐺𝑚) ∈ ω)
7253, 71syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → suc (𝐺𝑚) ∈ ω)
7372, 19jctil 312 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω))
74 0zd 9265 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → 0 ∈ ℤ)
75 id 19 . . . . . . . . . 10 ((𝐺𝑚) ∈ ω → (𝐺𝑚) ∈ ω)
7674, 17, 75frec2uzsucd 10401 . . . . . . . . 9 ((𝐺𝑚) ∈ ω → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7753, 76syl 14 . . . . . . . 8 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = ((𝐺‘(𝐺𝑚)) + 1))
7852eleq2i 2244 . . . . . . . . . . 11 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
7978biimpi 120 . . . . . . . . . 10 (𝑚 ∈ ℕ0𝑚 ∈ (ℤ‘0))
80 f1ocnvfv2 5779 . . . . . . . . . 10 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ 𝑚 ∈ (ℤ‘0)) → (𝐺‘(𝐺𝑚)) = 𝑚)
8119, 79, 80sylancr 414 . . . . . . . . 9 (𝑚 ∈ ℕ0 → (𝐺‘(𝐺𝑚)) = 𝑚)
8281oveq1d 5890 . . . . . . . 8 (𝑚 ∈ ℕ0 → ((𝐺‘(𝐺𝑚)) + 1) = (𝑚 + 1))
8377, 82eqtrd 2210 . . . . . . 7 (𝑚 ∈ ℕ0 → (𝐺‘suc (𝐺𝑚)) = (𝑚 + 1))
84 f1ocnvfv 5780 . . . . . . 7 ((𝐺:ω–1-1-onto→(ℤ‘0) ∧ suc (𝐺𝑚) ∈ ω) → ((𝐺‘suc (𝐺𝑚)) = (𝑚 + 1) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚)))
8573, 83, 84sylc 62 . . . . . 6 (𝑚 ∈ ℕ0 → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
8685adantr 276 . . . . 5 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = suc (𝐺𝑚))
87 df-suc 4372 . . . . 5 suc (𝐺𝑚) = ((𝐺𝑚) ∪ {(𝐺𝑚)})
8886, 87eqtrdi 2226 . . . 4 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (𝐺‘(𝑚 + 1)) = ((𝐺𝑚) ∪ {(𝐺𝑚)}))
8961, 70, 883brtr4d 4036 . . 3 ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (𝐺𝑚)) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1)))
9089ex 115 . 2 (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (𝐺𝑚) → (1...(𝑚 + 1)) ≈ (𝐺‘(𝑚 + 1))))
913, 6, 9, 12, 26, 90nn0ind 9367 1 (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (𝐺𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wal 1351   = wceq 1353  wtru 1354  wcel 2148  Vcvv 2738  cun 3128  cin 3129  c0 3423  {csn 3593   class class class wbr 4004  cmpt 4065  Ord word 4363  suc csuc 4366  ωcom 4590  ccnv 4626   Fn wfn 5212  1-1-ontowf1o 5216  cfv 5217  (class class class)co 5875  freccfrec 6391  cen 6738  0cc0 7811  1c1 7812   + caddc 7814  cmin 8128  0cn0 9176  cz 9253  cuz 9528  ...cfz 10008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-recs 6306  df-frec 6392  df-1o 6417  df-er 6535  df-en 6741  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009
This theorem is referenced by:  frecfzen2  10427  hashfz1  10763
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