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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 5904 |
. . . . 5
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2 | 1 | eleq1d 2258 |
. . . 4
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3 | 2 | imbi2d 230 |
. . 3
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4 | oveq2 5904 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | eleq1d 2258 |
. . . 4
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6 | oveq2 5904 |
. . . . 5
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7 | 6 | eleq1d 2258 |
. . . 4
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8 | oveq2 5904 |
. . . . 5
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9 | 8 | eleq1d 2258 |
. . . 4
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10 | nna0 6499 |
. . . . . 6
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11 | 10 | eleq1d 2258 |
. . . . 5
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12 | 11 | ibir 177 |
. . . 4
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13 | peano2 4612 |
. . . . . 6
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14 | nnasuc 6501 |
. . . . . . 7
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15 | 14 | eleq1d 2258 |
. . . . . 6
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16 | 13, 15 | imbitrrid 156 |
. . . . 5
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17 | 16 | expcom 116 |
. . . 4
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18 | 5, 7, 9, 12, 17 | finds2 4618 |
. . 3
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19 | 3, 18 | vtoclga 2818 |
. 2
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20 | 19 | impcom 125 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-recs 6330 df-irdg 6395 df-oadd 6445 |
This theorem is referenced by: nnmcl 6506 nnacli 6507 nnaass 6510 nndi 6511 nndir 6515 nnaordi 6533 nnaord 6534 nnaword 6536 addclpi 7356 nnppipi 7372 archnqq 7446 addcmpblnq0 7472 addclnq0 7480 nnanq0 7487 distrnq0 7488 addassnq0lemcl 7490 prarloclemlt 7522 prarloclemlo 7523 prarloclem3 7526 omgadd 10814 hashunlem 10816 hashun 10817 |
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