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Mirrors > Home > ILE Home > Th. List > nn0addcld | Unicode version |
Description: Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
nn0red.1 |
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nn0addcld.2 |
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Ref | Expression |
---|---|
nn0addcld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0red.1 |
. 2
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2 | nn0addcld.2 |
. 2
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3 | nn0addcl 9200 |
. 2
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4 | 1, 2, 3 | syl2anc 411 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 ax-sep 4118 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-i2m1 7907 ax-0id 7910 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-iota 5174 df-fv 5220 df-ov 5872 df-inn 8909 df-n0 9166 |
This theorem is referenced by: modsumfzodifsn 10382 expaddzap 10550 nn0opthlem1d 10684 nn0opthlem2d 10685 nn0opthd 10686 nn0opth2d 10687 bccl 10731 mertenslemi1 11527 pcpremul 12276 gzabssqcl 12362 4sqlem2 12370 mul4sq 12375 2sqlem8 14126 |
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