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Theorem 1tonninf 10106
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 5376 . . . 4 (𝐼‘1) = (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3 1nn0 8897 . . . . . 6 1 ∈ ℕ0
4 nn0xnn0 8948 . . . . . 6 (1 ∈ ℕ0 → 1 ∈ ℕ0*)
53, 4ax-mp 7 . . . . 5 1 ∈ ℕ0*
6 nn0nepnf 8952 . . . . . . 7 (1 ∈ ℕ0 → 1 ≠ +∞)
73, 6ax-mp 7 . . . . . 6 1 ≠ +∞
87necomi 2367 . . . . 5 +∞ ≠ 1
9 fvunsng 5568 . . . . 5 ((1 ∈ ℕ0* ∧ +∞ ≠ 1) → (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹𝐺)‘1))
105, 8, 9mp2an 420 . . . 4 (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1) = ((𝐹𝐺)‘1)
11 fxnn0nninf.g . . . . . . . 8 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
1211frechashgf1o 10094 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
13 f1ocnv 5336 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ℕ01-1-onto→ω)
1412, 13ax-mp 7 . . . . . 6 𝐺:ℕ01-1-onto→ω
15 f1of 5323 . . . . . 6 (𝐺:ℕ01-1-onto→ω → 𝐺:ℕ0⟶ω)
1614, 15ax-mp 7 . . . . 5 𝐺:ℕ0⟶ω
17 fvco3 5446 . . . . 5 ((𝐺:ℕ0⟶ω ∧ 1 ∈ ℕ0) → ((𝐹𝐺)‘1) = (𝐹‘(𝐺‘1)))
1816, 3, 17mp2an 420 . . . 4 ((𝐹𝐺)‘1) = (𝐹‘(𝐺‘1))
192, 10, 183eqtri 2139 . . 3 (𝐼‘1) = (𝐹‘(𝐺‘1))
20 df-1o 6267 . . . . . . 7 1o = suc ∅
2120fveq2i 5378 . . . . . 6 (𝐺‘1o) = (𝐺‘suc ∅)
22 0zd 8970 . . . . . . . . 9 (⊤ → 0 ∈ ℤ)
23 peano1 4468 . . . . . . . . . 10 ∅ ∈ ω
2423a1i 9 . . . . . . . . 9 (⊤ → ∅ ∈ ω)
2522, 11, 24frec2uzsucd 10067 . . . . . . . 8 (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
2625mptru 1323 . . . . . . 7 (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
2722, 11frec2uz0d 10065 . . . . . . . . 9 (⊤ → (𝐺‘∅) = 0)
2827mptru 1323 . . . . . . . 8 (𝐺‘∅) = 0
2928oveq1i 5738 . . . . . . 7 ((𝐺‘∅) + 1) = (0 + 1)
3026, 29eqtri 2135 . . . . . 6 (𝐺‘suc ∅) = (0 + 1)
31 0p1e1 8744 . . . . . 6 (0 + 1) = 1
3221, 30, 313eqtri 2139 . . . . 5 (𝐺‘1o) = 1
33 1onn 6370 . . . . . 6 1o ∈ ω
34 f1ocnvfv 5634 . . . . . 6 ((𝐺:ω–1-1-onto→ℕ0 ∧ 1o ∈ ω) → ((𝐺‘1o) = 1 → (𝐺‘1) = 1o))
3512, 33, 34mp2an 420 . . . . 5 ((𝐺‘1o) = 1 → (𝐺‘1) = 1o)
3632, 35ax-mp 7 . . . 4 (𝐺‘1) = 1o
3736fveq2i 5378 . . 3 (𝐹‘(𝐺‘1)) = (𝐹‘1o)
38 eleq2 2178 . . . . . . 7 (𝑛 = 1o → (𝑖𝑛𝑖 ∈ 1o))
3938ifbid 3459 . . . . . 6 (𝑛 = 1o → if(𝑖𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
4039mpteq2dv 3979 . . . . 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 4467 . . . . . 6 ω ∈ V
4342mptex 5600 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)) ∈ V
4440, 41, 43fvmpt3i 5455 . . . 4 (1o ∈ ω → (𝐹‘1o) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)))
4533, 44ax-mp 7 . . 3 (𝐹‘1o) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
4619, 37, 453eqtri 2139 . 2 (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47 el1o 6288 . . . 4 (𝑖 ∈ 1o𝑖 = ∅)
48 ifbi 3458 . . . 4 ((𝑖 ∈ 1o𝑖 = ∅) → if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅))
4947, 48ax-mp 7 . . 3 if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅)
5049mpteq2i 3975 . 2 (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51 eqeq1 2121 . . . 4 (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
5251ifbid 3459 . . 3 (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
5352cbvmptv 3984 . 2 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
5446, 50, 533eqtri 2139 1 (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1314  wtru 1315  wcel 1463  wne 2282  cun 3035  c0 3329  ifcif 3440  {csn 3493  cop 3496  cmpt 3949  suc csuc 4247  ωcom 4464   × cxp 4497  ccnv 4498  ccom 4503  wf 5077  1-1-ontowf1o 5080  cfv 5081  (class class class)co 5728  freccfrec 6241  1oc1o 6260  0cc0 7547  1c1 7548   + caddc 7550  +∞cpnf 7721  0cn0 8881  0*cxnn0 8944  cz 8958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-addcom 7645  ax-addass 7647  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-0id 7653  ax-rnegex 7654  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-ltadd 7661
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-recs 6156  df-frec 6242  df-1o 6267  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-inn 8631  df-n0 8882  df-xnn0 8945  df-z 8959  df-uz 9229
This theorem is referenced by: (None)
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