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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 5775 | . . . . 5 | |
2 | 1 | eleq1d 2206 | . . . 4 |
3 | 2 | imbi2d 229 | . . 3 |
4 | oveq2 5775 | . . . . 5 | |
5 | 4 | eleq1d 2206 | . . . 4 |
6 | 5 | imbi2d 229 | . . 3 |
7 | oveq2 5775 | . . . . 5 | |
8 | 7 | eleq1d 2206 | . . . 4 |
9 | 8 | imbi2d 229 | . . 3 |
10 | oveq2 5775 | . . . . 5 | |
11 | 10 | eleq1d 2206 | . . . 4 |
12 | 11 | imbi2d 229 | . . 3 |
13 | peano2nn 8725 | . . 3 | |
14 | peano2nn 8725 | . . . . . 6 | |
15 | nncn 8721 | . . . . . . . 8 | |
16 | nncn 8721 | . . . . . . . 8 | |
17 | ax-1cn 7706 | . . . . . . . . 9 | |
18 | addass 7743 | . . . . . . . . 9 | |
19 | 17, 18 | mp3an3 1304 | . . . . . . . 8 |
20 | 15, 16, 19 | syl2an 287 | . . . . . . 7 |
21 | 20 | eleq1d 2206 | . . . . . 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 8729 | . 2 |
26 | 25 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5767 cc 7611 c1 7614 caddc 7616 cn 8713 |
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 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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-addrcl 7710 ax-addass 7715 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 df-inn 8714 |
This theorem is referenced by: nnmulcl 8734 nn2ge 8746 nnaddcld 8761 nnnn0addcl 9000 nn0addcl 9005 9p1e10 9177 |
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