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Theorem 1tonninf 10413
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 5511 . . . 4 (𝐼‘1) = (((𝐹𝐺) ∪ {⟨+∞, (ω × {1o})⟩})‘1)
3 1nn0 9168 . . . . . 6 1 ∈ ℕ0
4 nn0xnn0 9219 . . . . . 6 (1 ∈ ℕ0 → 1 ∈ ℕ0*)
53, 4ax-mp 5 . . . . 5 1 ∈ ℕ0*
6 nn0nepnf 9223 . . . . . . 7 (1 ∈ ℕ0 → 1 ≠ +∞)
73, 6ax-mp 5 . . . . . 6 1 ≠ +∞
87necomi 2432 . . . . 5 +∞ ≠ 1
9 fvunsng 5705 . . . . 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 10401 . . . . . . 7 𝐺:ω–1-1-onto→ℕ0
13 f1ocnv 5469 . . . . . . 7 (𝐺:ω–1-1-onto→ℕ0𝐺:ℕ01-1-onto→ω)
1412, 13ax-mp 5 . . . . . 6 𝐺:ℕ01-1-onto→ω
15 f1of 5456 . . . . . 6 (𝐺:ℕ01-1-onto→ω → 𝐺:ℕ0⟶ω)
1614, 15ax-mp 5 . . . . 5 𝐺:ℕ0⟶ω
17 fvco3 5582 . . . . 5 ((𝐺:ℕ0⟶ω ∧ 1 ∈ ℕ0) → ((𝐹𝐺)‘1) = (𝐹‘(𝐺‘1)))
1816, 3, 17mp2an 426 . . . 4 ((𝐹𝐺)‘1) = (𝐹‘(𝐺‘1))
192, 10, 183eqtri 2202 . . 3 (𝐼‘1) = (𝐹‘(𝐺‘1))
20 df-1o 6410 . . . . . . 7 1o = suc ∅
2120fveq2i 5513 . . . . . 6 (𝐺‘1o) = (𝐺‘suc ∅)
22 0zd 9241 . . . . . . . . 9 (⊤ → 0 ∈ ℤ)
23 peano1 4589 . . . . . . . . . 10 ∅ ∈ ω
2423a1i 9 . . . . . . . . 9 (⊤ → ∅ ∈ ω)
2522, 11, 24frec2uzsucd 10374 . . . . . . . 8 (⊤ → (𝐺‘suc ∅) = ((𝐺‘∅) + 1))
2625mptru 1362 . . . . . . 7 (𝐺‘suc ∅) = ((𝐺‘∅) + 1)
2722, 11frec2uz0d 10372 . . . . . . . . 9 (⊤ → (𝐺‘∅) = 0)
2827mptru 1362 . . . . . . . 8 (𝐺‘∅) = 0
2928oveq1i 5878 . . . . . . 7 ((𝐺‘∅) + 1) = (0 + 1)
3026, 29eqtri 2198 . . . . . 6 (𝐺‘suc ∅) = (0 + 1)
31 0p1e1 9009 . . . . . 6 (0 + 1) = 1
3221, 30, 313eqtri 2202 . . . . 5 (𝐺‘1o) = 1
33 1onn 6514 . . . . . 6 1o ∈ ω
34 f1ocnvfv 5773 . . . . . 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 5513 . . 3 (𝐹‘(𝐺‘1)) = (𝐹‘1o)
38 eleq2 2241 . . . . . . 7 (𝑛 = 1o → (𝑖𝑛𝑖 ∈ 1o))
3938ifbid 3555 . . . . . 6 (𝑛 = 1o → if(𝑖𝑛, 1o, ∅) = if(𝑖 ∈ 1o, 1o, ∅))
4039mpteq2dv 4091 . . . . 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 4588 . . . . . 6 ω ∈ V
4342mptex 5737 . . . . 5 (𝑖 ∈ ω ↦ if(𝑖𝑛, 1o, ∅)) ∈ V
4440, 41, 43fvmpt3i 5591 . . . 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 2202 . 2 (𝐼‘1) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅))
47 el1o 6431 . . . 4 (𝑖 ∈ 1o𝑖 = ∅)
48 ifbi 3554 . . . 4 ((𝑖 ∈ 1o𝑖 = ∅) → if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅))
4947, 48ax-mp 5 . . 3 if(𝑖 ∈ 1o, 1o, ∅) = if(𝑖 = ∅, 1o, ∅)
5049mpteq2i 4087 . 2 (𝑖 ∈ ω ↦ if(𝑖 ∈ 1o, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅))
51 eqeq1 2184 . . . 4 (𝑖 = 𝑥 → (𝑖 = ∅ ↔ 𝑥 = ∅))
5251ifbid 3555 . . 3 (𝑖 = 𝑥 → if(𝑖 = ∅, 1o, ∅) = if(𝑥 = ∅, 1o, ∅))
5352cbvmptv 4096 . 2 (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1o, ∅)) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
5446, 50, 533eqtri 2202 1 (𝐼‘1) = (𝑥 ∈ ω ↦ if(𝑥 = ∅, 1o, ∅))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1353  wtru 1354  wcel 2148  wne 2347  cun 3127  c0 3422  ifcif 3534  {csn 3591  cop 3594  cmpt 4061  suc csuc 4361  ωcom 4585   × cxp 4620  ccnv 4621  ccom 4626  wf 5207  1-1-ontowf1o 5210  cfv 5211  (class class class)co 5868  freccfrec 6384  1oc1o 6403  0cc0 7789  1c1 7790   + caddc 7792  +∞cpnf 7966  0cn0 9152  0*cxnn0 9215  cz 9229
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-recs 6299  df-frec 6385  df-1o 6410  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-n0 9153  df-xnn0 9216  df-z 9230  df-uz 9505
This theorem is referenced by: (None)
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