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Mirrors > Home > ILE Home > Th. List > nnacl | Unicode version |
Description: Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnacl |
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 | oveq2 5775 | . . . . 5 | |
7 | 6 | eleq1d 2206 | . . . 4 |
8 | oveq2 5775 | . . . . 5 | |
9 | 8 | eleq1d 2206 | . . . 4 |
10 | nna0 6363 | . . . . . 6 | |
11 | 10 | eleq1d 2206 | . . . . 5 |
12 | 11 | ibir 176 | . . . 4 |
13 | peano2 4504 | . . . . . 6 | |
14 | nnasuc 6365 | . . . . . . 7 | |
15 | 14 | eleq1d 2206 | . . . . . 6 |
16 | 13, 15 | syl5ibr 155 | . . . . 5 |
17 | 16 | expcom 115 | . . . 4 |
18 | 5, 7, 9, 12, 17 | finds2 4510 | . . 3 |
19 | 3, 18 | vtoclga 2747 | . 2 |
20 | 19 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 c0 3358 csuc 4282 com 4499 (class class class)co 5767 coa 6303 |
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-in1 603 ax-in2 604 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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-irdg 6260 df-oadd 6310 |
This theorem is referenced by: nnmcl 6370 nnacli 6371 nnaass 6374 nndi 6375 nndir 6379 nnaordi 6397 nnaord 6398 nnaword 6400 addclpi 7128 nnppipi 7144 archnqq 7218 addcmpblnq0 7244 addclnq0 7252 nnanq0 7259 distrnq0 7260 addassnq0lemcl 7262 prarloclemlt 7294 prarloclemlo 7295 prarloclem3 7298 omgadd 10541 hashunlem 10543 hashun 10544 |
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