nn0addcld - Intuitionistic Logic Explorer

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Theorem nn0addcld 9633

Description: Closure of addition of nonnegative integers, inference form. (Contributed by Mario Carneiro, 27-May-2016.)

**Hypotheses**
Ref	Expression
nn0red.1
nn0addcld.2

**Assertion**
Ref	Expression
nn0addcld

**Proof of Theorem nn0addcld**
Step	Hyp	Ref	Expression
1		nn0red.1	. 2
2		nn0addcld.2	. 2
3		nn0addcl 9609	. 2 $/\$
4	1, 2, 3	syl2anc 415	1

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4

wcel 2209 (class class class)co 6088

caddc 8188

cn0 9574

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-ext 2220 ax-sep 4251 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-addass 8287 ax-i2m1 8290 ax-0id 8293

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4132 df-iota 5340 df-fv 5388 df-ov 6091 df-inn 9314 df-n0 9575

This theorem is used by: modsumfzodifsn 10853 expaddzap 11040 nn0opthlem1d 11179 nn0opthlem2d 11180 nn0opthd 11181 nn0opth2d 11182 bccl 11226 ccatfvalfi 11381 ccatcl 11382 ccatalpha 11402 swrdccat2 11464 mertenslemi1 12326 bitsmod 12747 bitsinv1lem 12752 pcpremul 13100 gzabssqcl 13188 4sqlem2 13196 mul4sq 13201 4sqlemsdc 13207 4sqlem12 13209 4sqlem14 13211 4sqlem16 13213 plymullem 15993 chtublem 16312 lgseisenlem2 16412 2sqlem8 16464 vtxdgfif 16756 clwwlknccat 16886