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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 5376 | . . . 4 ⊢ (𝐼‘0) = (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘0) |
3 | 0xnn0 8950 | . . . . 5 ⊢ 0 ∈ ℕ0* | |
4 | 0nn0 8896 | . . . . . . 7 ⊢ 0 ∈ ℕ0 | |
5 | nn0nepnf 8952 | . . . . . . 7 ⊢ (0 ∈ ℕ0 → 0 ≠ +∞) | |
6 | 4, 5 | ax-mp 7 | . . . . . 6 ⊢ 0 ≠ +∞ |
7 | 6 | necomi 2367 | . . . . 5 ⊢ +∞ ≠ 0 |
8 | fvunsng 5568 | . . . . 5 ⊢ ((0 ∈ ℕ0* ∧ +∞ ≠ 0) → (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘0) = ((𝐹 ∘ ◡𝐺)‘0)) | |
9 | 3, 7, 8 | mp2an 420 | . . . 4 ⊢ (((𝐹 ∘ ◡𝐺) ∪ {〈+∞, (ω × {1o})〉})‘0) = ((𝐹 ∘ ◡𝐺)‘0) |
10 | fxnn0nninf.g | . . . . . . . 8 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
11 | 10 | frechashgf1o 10094 | . . . . . . 7 ⊢ 𝐺:ω–1-1-onto→ℕ0 |
12 | f1ocnv 5336 | . . . . . . 7 ⊢ (𝐺:ω–1-1-onto→ℕ0 → ◡𝐺:ℕ0–1-1-onto→ω) | |
13 | 11, 12 | ax-mp 7 | . . . . . 6 ⊢ ◡𝐺:ℕ0–1-1-onto→ω |
14 | f1of 5323 | . . . . . 6 ⊢ (◡𝐺:ℕ0–1-1-onto→ω → ◡𝐺:ℕ0⟶ω) | |
15 | 13, 14 | ax-mp 7 | . . . . 5 ⊢ ◡𝐺:ℕ0⟶ω |
16 | fvco3 5446 | . . . . 5 ⊢ ((◡𝐺:ℕ0⟶ω ∧ 0 ∈ ℕ0) → ((𝐹 ∘ ◡𝐺)‘0) = (𝐹‘(◡𝐺‘0))) | |
17 | 15, 4, 16 | mp2an 420 | . . . 4 ⊢ ((𝐹 ∘ ◡𝐺)‘0) = (𝐹‘(◡𝐺‘0)) |
18 | 2, 9, 17 | 3eqtri 2139 | . . 3 ⊢ (𝐼‘0) = (𝐹‘(◡𝐺‘0)) |
19 | 0zd 8970 | . . . . . . 7 ⊢ (⊤ → 0 ∈ ℤ) | |
20 | 19, 10 | frec2uz0d 10065 | . . . . . 6 ⊢ (⊤ → (𝐺‘∅) = 0) |
21 | 20 | mptru 1323 | . . . . 5 ⊢ (𝐺‘∅) = 0 |
22 | peano1 4468 | . . . . . 6 ⊢ ∅ ∈ ω | |
23 | f1ocnvfv 5634 | . . . . . 6 ⊢ ((𝐺:ω–1-1-onto→ℕ0 ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅)) | |
24 | 11, 22, 23 | mp2an 420 | . . . . 5 ⊢ ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅) |
25 | 21, 24 | ax-mp 7 | . . . 4 ⊢ (◡𝐺‘0) = ∅ |
26 | 25 | fveq2i 5378 | . . 3 ⊢ (𝐹‘(◡𝐺‘0)) = (𝐹‘∅) |
27 | eleq2 2178 | . . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅)) | |
28 | 27 | ifbid 3459 | . . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ ∅, 1o, ∅)) |
29 | 28 | mpteq2dv 3979 | . . . . 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 4467 | . . . . . 6 ⊢ ω ∈ V | |
32 | 31 | mptex 5600 | . . . . 5 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ V |
33 | 29, 30, 32 | fvmpt3i 5455 | . . . 4 ⊢ (∅ ∈ ω → (𝐹‘∅) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))) |
34 | 22, 33 | ax-mp 7 | . . 3 ⊢ (𝐹‘∅) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) |
35 | 18, 26, 34 | 3eqtri 2139 | . 2 ⊢ (𝐼‘0) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) |
36 | noel 3333 | . . . 4 ⊢ ¬ 𝑖 ∈ ∅ | |
37 | 36 | iffalsei 3449 | . . 3 ⊢ if(𝑖 ∈ ∅, 1o, ∅) = ∅ |
38 | 37 | mpteq2i 3975 | . 2 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) = (𝑖 ∈ ω ↦ ∅) |
39 | eqidd 2116 | . . 3 ⊢ (𝑖 = 𝑥 → ∅ = ∅) | |
40 | 39 | cbvmptv 3984 | . 2 ⊢ (𝑖 ∈ ω ↦ ∅) = (𝑥 ∈ ω ↦ ∅) |
41 | 35, 38, 40 | 3eqtri 2139 | 1 ⊢ (𝐼‘0) = (𝑥 ∈ ω ↦ ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ⊤wtru 1315 ∈ wcel 1463 ≠ wne 2282 ∪ cun 3035 ∅c0 3329 ifcif 3440 {csn 3493 〈cop 3496 ↦ cmpt 3949 ωcom 4464 × cxp 4497 ◡ccnv 4498 ∘ ccom 4503 ⟶wf 5077 –1-1-onto→wf1o 5080 ‘cfv 5081 (class class class)co 5728 freccfrec 6241 1oc1o 6260 0cc0 7547 1c1 7548 + caddc 7550 +∞cpnf 7721 ℕ0cn0 8881 ℕ0*cxnn0 8944 ℤcz 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-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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