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Mirrors > Home > ILE Home > Th. List > infnninf | Unicode version |
Description: The point at infinity in ℕ∞ is the constant sequence equal to . Note that with our encoding of functions, that constant function can also be expressed as , as fconstmpt 4645 shows. (Contributed by Jim Kingdon, 14-Jul-2022.) Use maps-to notation. (Revised by BJ, 10-Aug-2024.) |
Ref | Expression |
---|---|
infnninf | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2o 6401 | . . . . . 6 | |
2 | 1 | a1i 9 | . . . . 5 |
3 | 2 | fmpttd 5634 | . . . 4 |
4 | 3 | mptru 1351 | . . 3 |
5 | 2on 6384 | . . . 4 | |
6 | omex 4564 | . . . 4 | |
7 | elmapg 6618 | . . . 4 | |
8 | 5, 6, 7 | mp2an 423 | . . 3 |
9 | 4, 8 | mpbir 145 | . 2 |
10 | peano2 4566 | . . . . . 6 | |
11 | eqidd 2165 | . . . . . . 7 | |
12 | eqid 2164 | . . . . . . 7 | |
13 | 1oex 6383 | . . . . . . 7 | |
14 | 11, 12, 13 | fvmpt 5557 | . . . . . 6 |
15 | 10, 14 | syl 14 | . . . . 5 |
16 | eqidd 2165 | . . . . . 6 | |
17 | 16, 12, 13 | fvmpt 5557 | . . . . 5 |
18 | 15, 17 | eqtr4d 2200 | . . . 4 |
19 | eqimss 3191 | . . . 4 | |
20 | 18, 19 | syl 14 | . . 3 |
21 | 20 | rgen 2517 | . 2 |
22 | fveq1 5479 | . . . . 5 | |
23 | fveq1 5479 | . . . . 5 | |
24 | 22, 23 | sseq12d 3168 | . . . 4 |
25 | 24 | ralbidv 2464 | . . 3 |
26 | df-nninf 7076 | . . 3 ℕ∞ | |
27 | 25, 26 | elrab2 2880 | . 2 ℕ∞ |
28 | 9, 21, 27 | mpbir2an 931 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1342 wtru 1343 wcel 2135 wral 2442 cvv 2721 wss 3111 cmpt 4037 con0 4335 csuc 4337 com 4561 wf 5178 cfv 5182 (class class class)co 5836 c1o 6368 c2o 6369 cmap 6605 ℕ∞xnninf 7075 |
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-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 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-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 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-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1o 6375 df-2o 6376 df-map 6607 df-nninf 7076 |
This theorem is referenced by: nnnninf2 7082 nninffeq 13741 |
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