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Mirrors > Home > ILE Home > Th. List > nnaddcl | Unicode version |
Description: Closure of addition of positive integers, proved by induction on the second addend. (Contributed by NM, 12-Jan-1997.) |
Ref | Expression |
---|---|
nnaddcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5750 | . . . . 5 | |
2 | 1 | eleq1d 2186 | . . . 4 |
3 | 2 | imbi2d 229 | . . 3 |
4 | oveq2 5750 | . . . . 5 | |
5 | 4 | eleq1d 2186 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | oveq2 5750 | . . . . 5 | |
8 | 7 | eleq1d 2186 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | oveq2 5750 | . . . . 5 | |
11 | 10 | eleq1d 2186 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | peano2nn 8696 | . . 3 | |
14 | peano2nn 8696 | . . . . . 6 | |
15 | nncn 8692 | . . . . . . . 8 | |
16 | nncn 8692 | . . . . . . . 8 | |
17 | ax-1cn 7681 | . . . . . . . . 9 | |
18 | addass 7718 | . . . . . . . . 9 | |
19 | 17, 18 | mp3an3 1289 | . . . . . . . 8 |
20 | 15, 16, 19 | syl2an 287 | . . . . . . 7 |
21 | 20 | eleq1d 2186 | . . . . . 6 |
22 | 14, 21 | syl5ib 153 | . . . . 5 |
23 | 22 | expcom 115 | . . . 4 |
24 | 23 | a2d 26 | . . 3 |
25 | 3, 6, 9, 12, 13, 24 | nnind 8700 | . 2 |
26 | 25 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 (class class class)co 5742 cc 7586 c1 7589 caddc 7591 cn 8684 |
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-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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-addrcl 7685 ax-addass 7690 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-iota 5058 df-fv 5101 df-ov 5745 df-inn 8685 |
This theorem is referenced by: nnmulcl 8705 nn2ge 8717 nnaddcld 8732 nnnn0addcl 8965 nn0addcl 8970 9p1e10 9142 |
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