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Mirrors > Home > ILE Home > Th. List > 0tonninf | GIF version |
Description: The mapping of zero into ℕ∞ is the sequence of all zeroes. (Contributed by Jim Kingdon, 17-Jul-2022.) |
Ref | Expression |
---|---|
fxnn0nninf.g | ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
fxnn0nninf.f | ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) |
fxnn0nninf.i | ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) |
Ref | Expression |
---|---|
0tonninf | ⊢ (𝐼‘0) = (𝑥 ∈ ω ↦ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fxnn0nninf.i | . . . . 5 ⊢ 𝐼 = ((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉}) | |
2 | 1 | fveq1i 5481 | . . . 4 ⊢ (𝐼‘0) = (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘0) |
3 | 0xnn0 9174 | . . . . 5 ⊢ 0 ∈ ℕ0* | |
4 | 0nn0 9120 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | nn0nepnf 9176 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ 0 ≠ +∞ |
7 | 6 | necomi 2419 | . . . . 5 ⊢ +∞ ≠ 0 |
8 | fvunsng 5673 | . . . . 5 ⊢ ((0 ∈ ℕ0* ∧ +∞ ≠ 0) → (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘0) = ((𝐹 ∘ ◡𝐺)‘0)) | |
9 | 3, 7, 8 | mp2an 423 | . . . 4 ⊢ (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘0) = ((𝐹 ∘ ◡𝐺)‘0) |
10 | fxnn0nninf.g | . . . . . . . 8 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
11 | 10 | frechashgf1o 10353 | . . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0 |
12 | f1ocnv 5439 | . . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → ◡𝐺:ℕ0–1-1-onto→ω) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ ◡𝐺:ℕ0–1-1-onto→ω |
14 | f1of 5426 | . . . . . 6 ⊢ (◡𝐺:ℕ0–1-1-onto→ω → ◡𝐺:ℕ0⟶ω) | |
15 | 13, 14 | ax-mp 5 | . . . . 5 ⊢ ◡𝐺:ℕ0⟶ω |
16 | fvco3 5551 | . . . . 5 ⊢ ((◡𝐺:ℕ0⟶ω ∧ 0 ∈ ℕ0) → ((𝐹 ∘ ◡𝐺)‘0) = (𝐹‘(◡𝐺‘0))) | |
17 | 15, 4, 16 | mp2an 423 | . . . 4 ⊢ ((𝐹 ∘ ◡𝐺)‘0) = (𝐹‘(◡𝐺‘0)) |
18 | 2, 9, 17 | 3eqtri 2189 | . . 3 ⊢ (𝐼‘0) = (𝐹‘(◡𝐺‘0)) |
19 | 0zd 9194 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℤ) | |
20 | 19, 10 | frec2uz0d 10324 | . . . . . 6 ⊢ (⊤ → (𝐺‘∅) = 0) |
21 | 20 | mptru 1351 | . . . . 5 ⊢ (𝐺‘∅) = 0 |
22 | peano1 4565 | . . . . . 6 ⊢ ∅ ∈ ω | |
23 | f1ocnvfv 5741 | . . . . . 6 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅)) | |
24 | 11, 22, 23 | mp2an 423 | . . . . 5 ⊢ ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅) |
25 | 21, 24 | ax-mp 5 | . . . 4 ⊢ (◡𝐺‘0) = ∅ |
26 | 25 | fveq2i 5483 | . . 3 ⊢ (𝐹‘(◡𝐺‘0)) = (𝐹‘∅) |
27 | eleq2 2228 | . . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅)) | |
28 | 27 | ifbid 3536 | . . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ ∅, 1o, ∅)) |
29 | 28 | mpteq2dv 4067 | . . . . 5 ⊢ (𝑛 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))) |
30 | fxnn0nninf.f | . . . . 5 ⊢ 𝐹 = (𝑛 ∈ ω ↦ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) | |
31 | omex 4564 | . . . . . 6 ⊢ ω ∈ V | |
32 | 31 | mptex 5705 | . . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ V |
33 | 29, 30, 32 | fvmpt3i 5560 | . . . 4 ⊢ (∅ ∈ ω → (𝐹‘∅) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))) |
34 | 22, 33 | ax-mp 5 | . . 3 ⊢ (𝐹‘∅) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) |
35 | 18, 26, 34 | 3eqtri 2189 | . 2 ⊢ (𝐼‘0) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) |
36 | noel 3408 | . . . 4 ⊢ ¬ 𝑖 ∈ ∅ | |
37 | 36 | iffalsei 3524 | . . 3 ⊢ if(𝑖 ∈ ∅, 1o, ∅) = ∅ |
38 | 37 | mpteq2i 4063 | . 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) = (𝑖 ∈ ω ↦ ∅) |
39 | eqidd 2165 | . . 3 ⊢ (𝑖 = 𝑥 → ∅ = ∅) | |
40 | 39 | cbvmptv 4072 | . 2 ⊢ (𝑖 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅) |
41 | 35, 38, 40 | 3eqtri 2189 | 1 ⊢ (𝐼‘0) = (𝑥 ∈ ω ↦ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ⊤wtru 1343 ∈ wcel 2135 ≠ wne 2334 ∪ cun 3109 ∅c0 3404 ifcif 3515 {csn 3570 〈cop 3573 ↦ cmpt 4037 ωcom 4561 × cxp 4596 ◡ccnv 4597 ∘ ccom 4602 ⟶wf 5178 –1-1-onto→wf1o 5181 ‘cfv 5182 (class class class)co 5836 freccfrec 6349 1oc1o 6368 0cc0 7744 1c1 7745 + caddc 7747 +∞cpnf 7921 ℕ0cn0 9105 ℕ0*cxnn0 9168 ℤcz 9182 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-ltadd 7860 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-inn 8849 df-n0 9106 df-xnn0 9169 df-z 9183 df-uz 9458 |
This theorem is referenced by: (None) |
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