Mathbox for Jim Kingdon |
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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfall | Unicode version |
Description: Given a decidable predicate on ℕ∞, showing it holds for natural numbers and the point at infinity suffices to show it holds everywhere. The sense in which is a decidable predicate is that it assigns a value of either or (which can be thought of as false and true) to every element of ℕ∞. Lemma 3.5 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 1-Aug-2022.) |
Ref | Expression |
---|---|
nninfall.q | ℕ∞ |
nninfall.inf | |
nninfall.n |
Ref | Expression |
---|---|
nninfall | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 6329 | . . . . 5 | |
2 | 1 | nesymi 2354 | . . . 4 |
3 | simplr 519 | . . . . . . . . . . 11 ℕ∞ ℕ∞ | |
4 | nninff 13198 | . . . . . . . . . . . 12 ℕ∞ | |
5 | 4 | ffnd 5273 | . . . . . . . . . . 11 ℕ∞ |
6 | 3, 5 | syl 14 | . . . . . . . . . 10 ℕ∞ |
7 | nninfall.q | . . . . . . . . . . . . . . 15 ℕ∞ | |
8 | 7 | ad2antrr 479 | . . . . . . . . . . . . . 14 ℕ∞ ℕ∞ |
9 | nninfall.inf | . . . . . . . . . . . . . . 15 | |
10 | 9 | ad2antrr 479 | . . . . . . . . . . . . . 14 ℕ∞ |
11 | nninfall.n | . . . . . . . . . . . . . . 15 | |
12 | 11 | ad2antrr 479 | . . . . . . . . . . . . . 14 ℕ∞ |
13 | simpr 109 | . . . . . . . . . . . . . 14 ℕ∞ | |
14 | 8, 10, 12, 3, 13 | nninfalllem1 13203 | . . . . . . . . . . . . 13 ℕ∞ |
15 | eqeq1 2146 | . . . . . . . . . . . . . . 15 | |
16 | 15 | ralrn 5558 | . . . . . . . . . . . . . 14 |
17 | 3, 5, 16 | 3syl 17 | . . . . . . . . . . . . 13 ℕ∞ |
18 | 14, 17 | mpbird 166 | . . . . . . . . . . . 12 ℕ∞ |
19 | peano1 4508 | . . . . . . . . . . . . . . . 16 | |
20 | elex2 2702 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | ax-mp 5 | . . . . . . . . . . . . . . 15 |
22 | fdm 5278 | . . . . . . . . . . . . . . . . 17 | |
23 | 22 | eleq2d 2209 | . . . . . . . . . . . . . . . 16 |
24 | 23 | exbidv 1797 | . . . . . . . . . . . . . . 15 |
25 | 21, 24 | mpbiri 167 | . . . . . . . . . . . . . 14 |
26 | dmmrnm 4758 | . . . . . . . . . . . . . . 15 | |
27 | eqsnm 3682 | . . . . . . . . . . . . . . 15 | |
28 | 26, 27 | sylbi 120 | . . . . . . . . . . . . . 14 |
29 | 25, 28 | syl 14 | . . . . . . . . . . . . 13 |
30 | 3, 4, 29 | 3syl 17 | . . . . . . . . . . . 12 ℕ∞ |
31 | 18, 30 | mpbird 166 | . . . . . . . . . . 11 ℕ∞ |
32 | eqimss 3151 | . . . . . . . . . . 11 | |
33 | 31, 32 | syl 14 | . . . . . . . . . 10 ℕ∞ |
34 | df-f 5127 | . . . . . . . . . 10 | |
35 | 6, 33, 34 | sylanbrc 413 | . . . . . . . . 9 ℕ∞ |
36 | 1onn 6416 | . . . . . . . . . 10 | |
37 | fconst2g 5635 | . . . . . . . . . 10 | |
38 | 36, 37 | ax-mp 5 | . . . . . . . . 9 |
39 | 35, 38 | sylib 121 | . . . . . . . 8 ℕ∞ |
40 | fconstmpt 4586 | . . . . . . . 8 | |
41 | 39, 40 | syl6eq 2188 | . . . . . . 7 ℕ∞ |
42 | 41 | fveq2d 5425 | . . . . . 6 ℕ∞ |
43 | 42, 13, 10 | 3eqtr3d 2180 | . . . . 5 ℕ∞ |
44 | 43 | ex 114 | . . . 4 ℕ∞ |
45 | 2, 44 | mtoi 653 | . . 3 ℕ∞ |
46 | elmapi 6564 | . . . . . . 7 ℕ∞ ℕ∞ | |
47 | 7, 46 | syl 14 | . . . . . 6 ℕ∞ |
48 | 47 | ffvelrnda 5555 | . . . . 5 ℕ∞ |
49 | elpri 3550 | . . . . . 6 | |
50 | df2o3 6327 | . . . . . 6 | |
51 | 49, 50 | eleq2s 2234 | . . . . 5 |
52 | 48, 51 | syl 14 | . . . 4 ℕ∞ |
53 | 52 | orcomd 718 | . . 3 ℕ∞ |
54 | 45, 53 | ecased 1327 | . 2 ℕ∞ |
55 | 54 | ralrimiva 2505 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wral 2416 wss 3071 c0 3363 cif 3474 csn 3527 cpr 3528 cmpt 3989 com 4504 cxp 4537 cdm 4539 crn 4540 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 c1o 6306 c2o 6307 cmap 6542 ℕ∞xnninf 7005 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1o 6313 df-2o 6314 df-map 6544 df-nninf 7007 |
This theorem is referenced by: nninfsel 13213 nninffeq 13216 |
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