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Mirrors > Home > ILE Home > Th. List > nnaddcld | Unicode version |
Description: Closure of addition of positive integers. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nnge1d.1 | |
nnmulcld.2 |
Ref | Expression |
---|---|
nnaddcld |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnge1d.1 | . 2 | |
2 | nnmulcld.2 | . 2 | |
3 | nnaddcl 8868 | . 2 | |
4 | 1, 2, 3 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 2135 (class class class)co 5836 caddc 7747 cn 8848 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-addrcl 7841 ax-addass 7846 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-inn 8849 |
This theorem is referenced by: pythagtriplem4 12179 pythagtriplem6 12181 pythagtriplem7 12182 pythagtriplem11 12185 pythagtriplem12 12186 pythagtriplem13 12187 pythagtriplem14 12188 pythagtriplem15 12189 pythagtriplem16 12190 |
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