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 6297 | . . . . 5 | |
2 | 1 | nesymi 2331 | . . . 4 |
3 | simplr 504 | . . . . . . . . . . 11 ℕ∞ ℕ∞ | |
4 | nninff 13094 | . . . . . . . . . . . 12 ℕ∞ | |
5 | 4 | ffnd 5243 | . . . . . . . . . . 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 13099 | . . . . . . . . . . . . 13 ℕ∞ |
15 | eqeq1 2124 | . . . . . . . . . . . . . . 15 | |
16 | 15 | ralrn 5526 | . . . . . . . . . . . . . 14 |
17 | 3, 5, 16 | 3syl 17 | . . . . . . . . . . . . 13 ℕ∞ |
18 | 14, 17 | mpbird 166 | . . . . . . . . . . . 12 ℕ∞ |
19 | peano1 4478 | . . . . . . . . . . . . . . . 16 | |
20 | elex2 2676 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | ax-mp 5 | . . . . . . . . . . . . . . 15 |
22 | fdm 5248 | . . . . . . . . . . . . . . . . 17 | |
23 | 22 | eleq2d 2187 | . . . . . . . . . . . . . . . 16 |
24 | 23 | exbidv 1781 | . . . . . . . . . . . . . . 15 |
25 | 21, 24 | mpbiri 167 | . . . . . . . . . . . . . 14 |
26 | dmmrnm 4728 | . . . . . . . . . . . . . . 15 | |
27 | eqsnm 3652 | . . . . . . . . . . . . . . 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 3121 | . . . . . . . . . . 11 | |
33 | 31, 32 | syl 14 | . . . . . . . . . 10 ℕ∞ |
34 | df-f 5097 | . . . . . . . . . 10 | |
35 | 6, 33, 34 | sylanbrc 413 | . . . . . . . . 9 ℕ∞ |
36 | 1onn 6384 | . . . . . . . . . 10 | |
37 | fconst2g 5603 | . . . . . . . . . 10 | |
38 | 36, 37 | ax-mp 5 | . . . . . . . . 9 |
39 | 35, 38 | sylib 121 | . . . . . . . 8 ℕ∞ |
40 | fconstmpt 4556 | . . . . . . . 8 | |
41 | 39, 40 | syl6eq 2166 | . . . . . . 7 ℕ∞ |
42 | 41 | fveq2d 5393 | . . . . . 6 ℕ∞ |
43 | 42, 13, 10 | 3eqtr3d 2158 | . . . . 5 ℕ∞ |
44 | 43 | ex 114 | . . . 4 ℕ∞ |
45 | 2, 44 | mtoi 638 | . . 3 ℕ∞ |
46 | elmapi 6532 | . . . . . . 7 ℕ∞ ℕ∞ | |
47 | 7, 46 | syl 14 | . . . . . 6 ℕ∞ |
48 | 47 | ffvelrnda 5523 | . . . . 5 ℕ∞ |
49 | elpri 3520 | . . . . . 6 | |
50 | df2o3 6295 | . . . . . 6 | |
51 | 49, 50 | eleq2s 2212 | . . . . 5 |
52 | 48, 51 | syl 14 | . . . 4 ℕ∞ |
53 | 52 | orcomd 703 | . . 3 ℕ∞ |
54 | 45, 53 | ecased 1312 | . 2 ℕ∞ |
55 | 54 | ralrimiva 2482 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wceq 1316 wex 1453 wcel 1465 wral 2393 wss 3041 c0 3333 cif 3444 csn 3497 cpr 3498 cmpt 3959 com 4474 cxp 4507 cdm 4509 crn 4510 wfn 5088 wf 5089 cfv 5093 (class class class)co 5742 c1o 6274 c2o 6275 cmap 6510 ℕ∞xnninf 6973 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1o 6281 df-2o 6282 df-map 6512 df-nninf 6975 |
This theorem is referenced by: nninfsel 13109 nninffeq 13112 |
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