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 6376 | . . . . 5 | |
2 | 1 | nesymi 2373 | . . . 4 |
3 | simplr 520 | . . . . . . . . . . 11 ℕ∞ ℕ∞ | |
4 | nninff 7061 | . . . . . . . . . . . 12 ℕ∞ | |
5 | 4 | ffnd 5319 | . . . . . . . . . . 11 ℕ∞ |
6 | 3, 5 | syl 14 | . . . . . . . . . 10 ℕ∞ |
7 | nninfall.q | . . . . . . . . . . . . . . 15 ℕ∞ | |
8 | 7 | ad2antrr 480 | . . . . . . . . . . . . . 14 ℕ∞ ℕ∞ |
9 | nninfall.inf | . . . . . . . . . . . . . . 15 | |
10 | 9 | ad2antrr 480 | . . . . . . . . . . . . . 14 ℕ∞ |
11 | nninfall.n | . . . . . . . . . . . . . . 15 | |
12 | 11 | ad2antrr 480 | . . . . . . . . . . . . . 14 ℕ∞ |
13 | simpr 109 | . . . . . . . . . . . . . 14 ℕ∞ | |
14 | 8, 10, 12, 3, 13 | nninfalllem1 13567 | . . . . . . . . . . . . 13 ℕ∞ |
15 | eqeq1 2164 | . . . . . . . . . . . . . . 15 | |
16 | 15 | ralrn 5604 | . . . . . . . . . . . . . 14 |
17 | 3, 5, 16 | 3syl 17 | . . . . . . . . . . . . 13 ℕ∞ |
18 | 14, 17 | mpbird 166 | . . . . . . . . . . . 12 ℕ∞ |
19 | peano1 4552 | . . . . . . . . . . . . . . . 16 | |
20 | elex2 2728 | . . . . . . . . . . . . . . . 16 | |
21 | 19, 20 | ax-mp 5 | . . . . . . . . . . . . . . 15 |
22 | fdm 5324 | . . . . . . . . . . . . . . . . 17 | |
23 | 22 | eleq2d 2227 | . . . . . . . . . . . . . . . 16 |
24 | 23 | exbidv 1805 | . . . . . . . . . . . . . . 15 |
25 | 21, 24 | mpbiri 167 | . . . . . . . . . . . . . 14 |
26 | dmmrnm 4804 | . . . . . . . . . . . . . . 15 | |
27 | eqsnm 3718 | . . . . . . . . . . . . . . 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 3182 | . . . . . . . . . . 11 | |
33 | 31, 32 | syl 14 | . . . . . . . . . 10 ℕ∞ |
34 | df-f 5173 | . . . . . . . . . 10 | |
35 | 6, 33, 34 | sylanbrc 414 | . . . . . . . . 9 ℕ∞ |
36 | 1onn 6464 | . . . . . . . . . 10 | |
37 | fconst2g 5681 | . . . . . . . . . 10 | |
38 | 36, 37 | ax-mp 5 | . . . . . . . . 9 |
39 | 35, 38 | sylib 121 | . . . . . . . 8 ℕ∞ |
40 | fconstmpt 4632 | . . . . . . . 8 | |
41 | 39, 40 | eqtrdi 2206 | . . . . . . 7 ℕ∞ |
42 | 41 | fveq2d 5471 | . . . . . 6 ℕ∞ |
43 | 42, 13, 10 | 3eqtr3d 2198 | . . . . 5 ℕ∞ |
44 | 43 | ex 114 | . . . 4 ℕ∞ |
45 | 2, 44 | mtoi 654 | . . 3 ℕ∞ |
46 | elmapi 6612 | . . . . . . 7 ℕ∞ ℕ∞ | |
47 | 7, 46 | syl 14 | . . . . . 6 ℕ∞ |
48 | 47 | ffvelrnda 5601 | . . . . 5 ℕ∞ |
49 | elpri 3583 | . . . . . 6 | |
50 | df2o3 6374 | . . . . . 6 | |
51 | 49, 50 | eleq2s 2252 | . . . . 5 |
52 | 48, 51 | syl 14 | . . . 4 ℕ∞ |
53 | 52 | orcomd 719 | . . 3 ℕ∞ |
54 | 45, 53 | ecased 1331 | . 2 ℕ∞ |
55 | 54 | ralrimiva 2530 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 wceq 1335 wex 1472 wcel 2128 wral 2435 wss 3102 c0 3394 cif 3505 csn 3560 cpr 3561 cmpt 4025 com 4548 cxp 4583 cdm 4585 crn 4586 wfn 5164 wf 5165 cfv 5169 (class class class)co 5821 c1o 6353 c2o 6354 cmap 6590 ℕ∞xnninf 7058 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1o 6360 df-2o 6361 df-map 6592 df-nninf 7059 |
This theorem is referenced by: nninfsel 13576 nninffeq 13579 |
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