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Mirrors > Home > ILE Home > Th. List > nnmcl | Unicode version |
Description: Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
nnmcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5714 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | 1 | eleq1d 2168 |
. . . 4
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3 | 2 | imbi2d 229 |
. . 3
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4 | oveq2 5714 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 4 | eleq1d 2168 |
. . . 4
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6 | oveq2 5714 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
7 | 6 | eleq1d 2168 |
. . . 4
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8 | oveq2 5714 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 8 | eleq1d 2168 |
. . . 4
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10 | nnm0 6301 |
. . . . 5
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11 | peano1 4446 |
. . . . 5
![]() ![]() ![]() ![]() | |
12 | 10, 11 | syl6eqel 2190 |
. . . 4
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13 | nnacl 6306 |
. . . . . . . 8
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14 | 13 | expcom 115 |
. . . . . . 7
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15 | 14 | adantr 272 |
. . . . . 6
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16 | nnmsuc 6303 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
17 | 16 | eleq1d 2168 |
. . . . . 6
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18 | 15, 17 | sylibrd 168 |
. . . . 5
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19 | 18 | expcom 115 |
. . . 4
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20 | 5, 7, 9, 12, 19 | finds2 4453 |
. . 3
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21 | 3, 20 | vtoclga 2707 |
. 2
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22 | 21 | impcom 124 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-nul 3994 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-nul 3311 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-tr 3967 df-id 4153 df-iord 4226 df-on 4228 df-suc 4231 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-recs 6132 df-irdg 6197 df-oadd 6247 df-omul 6248 |
This theorem is referenced by: nnmcli 6309 nndi 6312 nnmass 6313 nnmsucr 6314 nnmordi 6342 nnmord 6343 nnmword 6344 mulclpi 7037 enq0tr 7143 addcmpblnq0 7152 mulcmpblnq0 7153 mulcanenq0ec 7154 addclnq0 7160 mulclnq0 7161 nqpnq0nq 7162 distrnq0 7168 addassnq0lemcl 7170 addassnq0 7171 |
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