ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1tonninf GIF version

Theorem 1tonninf 10473
Description: The mapping of one into is a sequence which is a one followed by zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.)
Hypotheses
Ref Expression
fxnn0nninf.g 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
fxnn0nninf.f 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))
fxnn0nninf.i 𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
Assertion
Ref Expression
1tonninf (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
Distinct variable groups:   𝑖,𝑛   𝑥,𝑖
Allowed substitution hints:   𝐹(𝑥,𝑖,𝑛)   𝐺(𝑥,𝑖,𝑛)   𝐼(𝑥,𝑖,𝑛)

Proof of Theorem 1tonninf
StepHypRef Expression
1 fxnn0nninf.i . . . . 5 𝐼 = ((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})
21fveq1i 5535 . . . 4 (𝐼‘1) = (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3 1nn0 9223 . . . . . 6 1 ∈ ℕ0
4 nn0xnn0 9274 . . . . . 6 (1 ∈ ℕ0 → 1 ∈ ℕ0*)
53, 4ax-mp 5 . . . . 5 1 ∈ ℕ0*
6 nn0nepnf 9278 . . . . . . 7 (1 ∈ ℕ0 → 1 ≠ +∞)
73, 6ax-mp 5 . . . . . 6 1 ≠ +∞
87necomi 2445 . . . . 5 +∞ ≠ 1
9 fvunsng 5731 . . . . 5 ((1 ∈ ℕ0* ∧ +∞ ≠ 1) → (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹𝐺)‘1))
105, 8, 9mp2an 426 . . . 4 (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹𝐺)‘1)
11 fxnn0nninf.g . . . . . . . 8 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
1211frechashgf1o 10461 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
13 f1ocnv 5493 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ℕ01-1-onto→ω)
1412, 13ax-mp 5 . . . . . 6 𝐺:ℕ01-1-onto→ω
15 f1of 5480 . . . . . 6 (𝐺:ℕ01-1-onto→ω → 𝐺:ℕ0⟶ω)
1614, 15ax-mp 5 . . . . 5 𝐺:ℕ0⟶ω
17 fvco3 5608 . . . . 5 ((𝐺:ℕ0⟶ω ∧ 1 ∈ ℕ0) → ((𝐹𝐺)‘1) = (𝐹‘(𝐺‘1)))
1816, 3, 17mp2an 426 . . . 4 ((𝐹𝐺)‘1) = (𝐹‘(𝐺‘1))
192, 10, 183eqtri 2214 . . 3 (𝐼‘1) = (𝐹‘(𝐺‘1))
20 df-1o 6442 . . . . . . 7 1o = suc ∅
2120fveq2i 5537 . . . . . 6 (𝐺‘1o) = (𝐺‘suc ∅)
22 0zd 9296 . . . . . . . . 9 (⊤ → 0 ∈ ℤ)
23 peano1 4611 . . . . . . . . . 10 ∅ ∈ ω
2423a1i 9 . . . . . . . . 9 (⊤ → ∅ ∈ ω)
2522, 11, 24frec2uzsucd 10434 . . . . . . . 8 (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
2625mptru 1373 . . . . . . 7 (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
2722, 11frec2uz0d 10432 . . . . . . . . 9 (⊤ → (𝐺‘∅) = 0)
2827mptru 1373 . . . . . . . 8 (𝐺‘∅) = 0
2928oveq1i 5907 . . . . . . 7 ((𝐺‘∅) + 1) = (0 + 1)
3026, 29eqtri 2210 . . . . . 6 (𝐺‘suc ∅) = (0 + 1)
31 0p1e1 9064 . . . . . 6 (0 + 1) = 1
3221, 30, 313eqtri 2214 . . . . 5 (𝐺‘1o) = 1
33 1onn 6546 . . . . . 6 1o ∈ ω
34 f1ocnvfv 5801 . . . . . 6 ((𝐺:ω–1-1-onto→ℕ0 ∧ 1o ∈ ω) → ((𝐺‘1o) = 1 → (𝐺‘1) = 1o))
3512, 33, 34mp2an 426 . . . . 5 ((𝐺‘1o) = 1 → (𝐺‘1) = 1o)
3632, 35ax-mp 5 . . . 4 (𝐺‘1) = 1o
3736fveq2i 5537 . . 3 (𝐹‘(𝐺‘1)) = (𝐹‘1o)
38 eleq2 2253 . . . . . . 7 (𝑛 = 1o → (𝑖𝑛𝑖 ∈ 1o))
3938ifbid 3570 . . . . . 6 (𝑛 = 1o → if(𝑖𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
4039mpteq2dv 4109 . . . . 5 (𝑛 = 1o → (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)))
41 fxnn0nninf.f . . . . 5 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)))
42 omex 4610 . . . . . 6 ω ∈ V
4342mptex 5763 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)) ∈ V
4440, 41, 43fvmpt3i 5617 . . . 4 (1o ∈ ω → (𝐹‘1o) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)))
4533, 44ax-mp 5 . . 3 (𝐹‘1o) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
4619, 37, 453eqtri 2214 . 2 (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47 el1o 6463 . . . 4 (𝑖 ∈ 1o𝑖 = ∅)
48 ifbi 3569 . . . 4 ((𝑖 ∈ 1o𝑖 = ∅) → if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅))
4947, 48ax-mp 5 . . 3 if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅)
5049mpteq2i 4105 . 2 (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51 eqeq1 2196 . . . 4 (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
5251ifbid 3570 . . 3 (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
5352cbvmptv 4114 . 2 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
5446, 50, 533eqtri 2214 1 (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1364  wtru 1365  wcel 2160  wne 2360  cun 3142  c0 3437  ifcif 3549  {csn 3607  cop 3610  cmpt 4079  suc csuc 4383  ωcom 4607   × cxp 4642  ccnv 4643  ccom 4648  wf 5231  1-1-ontowf1o 5234  cfv 5235  (class class class)co 5897  freccfrec 6416  1oc1o 6435  0cc0 7842  1c1 7843   + caddc 7845  +∞cpnf 8020  0cn0 9207  0*cxnn0 9270  cz 9284
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-recs 6331  df-frec 6417  df-1o 6442  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-xnn0 9271  df-z 9285  df-uz 9560
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator