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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5790 |
. . . . 5
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2 | 1 | eleq1d 2209 |
. . . 4
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3 | 2 | imbi2d 229 |
. . 3
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4 | oveq2 5790 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | eleq1d 2209 |
. . . 4
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6 | oveq2 5790 |
. . . . 5
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7 | 6 | eleq1d 2209 |
. . . 4
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8 | oveq2 5790 |
. . . . 5
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9 | 8 | eleq1d 2209 |
. . . 4
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10 | nna0 6378 |
. . . . . 6
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11 | 10 | eleq1d 2209 |
. . . . 5
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12 | 11 | ibir 176 |
. . . 4
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13 | peano2 4517 |
. . . . . 6
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14 | nnasuc 6380 |
. . . . . . 7
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15 | 14 | eleq1d 2209 |
. . . . . 6
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16 | 13, 15 | syl5ibr 155 |
. . . . 5
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17 | 16 | expcom 115 |
. . . 4
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18 | 5, 7, 9, 12, 17 | finds2 4523 |
. . 3
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19 | 3, 18 | vtoclga 2755 |
. 2
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20 | 19 | impcom 124 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-oadd 6325 |
This theorem is referenced by: nnmcl 6385 nnacli 6386 nnaass 6389 nndi 6390 nndir 6394 nnaordi 6412 nnaord 6413 nnaword 6415 addclpi 7159 nnppipi 7175 archnqq 7249 addcmpblnq0 7275 addclnq0 7283 nnanq0 7290 distrnq0 7291 addassnq0lemcl 7293 prarloclemlt 7325 prarloclemlo 7326 prarloclem3 7329 omgadd 10580 hashunlem 10582 hashun 10583 |
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