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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5844 | . . . . 5 | |
2 | 1 | eleq1d 2233 | . . . 4 |
3 | 2 | imbi2d 229 | . . 3 |
4 | oveq2 5844 | . . . . 5 | |
5 | 4 | eleq1d 2233 | . . . 4 |
6 | oveq2 5844 | . . . . 5 | |
7 | 6 | eleq1d 2233 | . . . 4 |
8 | oveq2 5844 | . . . . 5 | |
9 | 8 | eleq1d 2233 | . . . 4 |
10 | nnm0 6434 | . . . . 5 | |
11 | peano1 4565 | . . . . 5 | |
12 | 10, 11 | eqeltrdi 2255 | . . . 4 |
13 | nnacl 6439 | . . . . . . . 8 | |
14 | 13 | expcom 115 | . . . . . . 7 |
15 | 14 | adantr 274 | . . . . . 6 |
16 | nnmsuc 6436 | . . . . . . 7 | |
17 | 16 | eleq1d 2233 | . . . . . 6 |
18 | 15, 17 | sylibrd 168 | . . . . 5 |
19 | 18 | expcom 115 | . . . 4 |
20 | 5, 7, 9, 12, 19 | finds2 4572 | . . 3 |
21 | 3, 20 | vtoclga 2787 | . 2 |
22 | 21 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 c0 3404 csuc 4337 com 4561 (class class class)co 5836 coa 6372 comu 6373 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-oadd 6379 df-omul 6380 |
This theorem is referenced by: nnmcli 6442 nndi 6445 nnmass 6446 nnmsucr 6447 nnmordi 6475 nnmord 6476 nnmword 6477 mulclpi 7260 enq0tr 7366 addcmpblnq0 7375 mulcmpblnq0 7376 mulcanenq0ec 7377 addclnq0 7383 mulclnq0 7384 nqpnq0nq 7385 distrnq0 7391 addassnq0lemcl 7393 addassnq0 7394 |
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