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Theorem infnninf 6784
Description: The point at infinity in (the constant sequence equal to 1𝑜). (Contributed by Jim Kingdon, 14-Jul-2022.)
Assertion
Ref Expression
infnninf (ω × {1𝑜}) ∈ ℕ

Proof of Theorem infnninf
Dummy variables 𝑓 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1oex 6171 . . . . . 6 1𝑜 ∈ V
21sucid 4235 . . . . 5 1𝑜 ∈ suc 1𝑜
3 df-2o 6164 . . . . 5 2𝑜 = suc 1𝑜
42, 3eleqtrri 2163 . . . 4 1𝑜 ∈ 2𝑜
54fconst6 5194 . . 3 (ω × {1𝑜}):ω⟶2𝑜
6 2onn 6260 . . . . 5 2𝑜 ∈ ω
76elexi 2631 . . . 4 2𝑜 ∈ V
8 omex 4398 . . . 4 ω ∈ V
97, 8elmap 6414 . . 3 ((ω × {1𝑜}) ∈ (2𝑜𝑚 ω) ↔ (ω × {1𝑜}):ω⟶2𝑜)
105, 9mpbir 144 . 2 (ω × {1𝑜}) ∈ (2𝑜𝑚 ω)
11 peano2 4400 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
121fvconst2 5495 . . . . . 6 (suc 𝑖 ∈ ω → ((ω × {1𝑜})‘suc 𝑖) = 1𝑜)
1311, 12syl 14 . . . . 5 (𝑖 ∈ ω → ((ω × {1𝑜})‘suc 𝑖) = 1𝑜)
141fvconst2 5495 . . . . 5 (𝑖 ∈ ω → ((ω × {1𝑜})‘𝑖) = 1𝑜)
1513, 14eqtr4d 2123 . . . 4 (𝑖 ∈ ω → ((ω × {1𝑜})‘suc 𝑖) = ((ω × {1𝑜})‘𝑖))
16 eqimss 3076 . . . 4 (((ω × {1𝑜})‘suc 𝑖) = ((ω × {1𝑜})‘𝑖) → ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖))
1715, 16syl 14 . . 3 (𝑖 ∈ ω → ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖))
1817rgen 2428 . 2 𝑖 ∈ ω ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)
19 fveq1 5288 . . . . 5 (𝑓 = (ω × {1𝑜}) → (𝑓‘suc 𝑖) = ((ω × {1𝑜})‘suc 𝑖))
20 fveq1 5288 . . . . 5 (𝑓 = (ω × {1𝑜}) → (𝑓𝑖) = ((ω × {1𝑜})‘𝑖))
2119, 20sseq12d 3053 . . . 4 (𝑓 = (ω × {1𝑜}) → ((𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)))
2221ralbidv 2380 . . 3 (𝑓 = (ω × {1𝑜}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)))
23 df-nninf 6770 . . 3 = {𝑓 ∈ (2𝑜𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
2422, 23elrab2 2772 . 2 ((ω × {1𝑜}) ∈ ℕ ↔ ((ω × {1𝑜}) ∈ (2𝑜𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1𝑜})‘suc 𝑖) ⊆ ((ω × {1𝑜})‘𝑖)))
2510, 18, 24mpbir2an 888 1 (ω × {1𝑜}) ∈ ℕ
Colors of variables: wff set class
Syntax hints:   = wceq 1289  wcel 1438  wral 2359  wss 2997  {csn 3441  suc csuc 4183  ωcom 4395   × cxp 4426  wf 4998  cfv 5002  (class class class)co 5634  1𝑜c1o 6156  2𝑜c2o 6157  𝑚 cmap 6385  xnninf 6768
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1o 6163  df-2o 6164  df-map 6387  df-nninf 6770
This theorem is referenced by:  fxnn0nninf  9809
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