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Mirrors > Home > ILE Home > Th. List > nnmord | Unicode version |
Description: Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
nnmord |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnmordi 6412 | . . . . . 6 | |
2 | 1 | ex 114 | . . . . 5 |
3 | 2 | com23 78 | . . . 4 |
4 | 3 | impd 252 | . . 3 |
5 | 4 | 3adant1 999 | . 2 |
6 | ne0i 3369 | . . . . . . . 8 | |
7 | nnm0r 6375 | . . . . . . . . . 10 | |
8 | oveq1 5781 | . . . . . . . . . . 11 | |
9 | 8 | eqeq1d 2148 | . . . . . . . . . 10 |
10 | 7, 9 | syl5ibrcom 156 | . . . . . . . . 9 |
11 | 10 | necon3d 2352 | . . . . . . . 8 |
12 | 6, 11 | syl5 32 | . . . . . . 7 |
13 | 12 | adantr 274 | . . . . . 6 |
14 | nn0eln0 4533 | . . . . . . 7 | |
15 | 14 | adantl 275 | . . . . . 6 |
16 | 13, 15 | sylibrd 168 | . . . . 5 |
17 | 16 | 3adant1 999 | . . . 4 |
18 | oveq2 5782 | . . . . . . . . . 10 | |
19 | 18 | a1i 9 | . . . . . . . . 9 |
20 | nnmordi 6412 | . . . . . . . . . 10 | |
21 | 20 | 3adantl2 1138 | . . . . . . . . 9 |
22 | 19, 21 | orim12d 775 | . . . . . . . 8 |
23 | 22 | con3d 620 | . . . . . . 7 |
24 | simpl3 986 | . . . . . . . . 9 | |
25 | simpl1 984 | . . . . . . . . 9 | |
26 | nnmcl 6377 | . . . . . . . . 9 | |
27 | 24, 25, 26 | syl2anc 408 | . . . . . . . 8 |
28 | simpl2 985 | . . . . . . . . 9 | |
29 | nnmcl 6377 | . . . . . . . . 9 | |
30 | 24, 28, 29 | syl2anc 408 | . . . . . . . 8 |
31 | nntri2 6390 | . . . . . . . 8 | |
32 | 27, 30, 31 | syl2anc 408 | . . . . . . 7 |
33 | nntri2 6390 | . . . . . . . 8 | |
34 | 25, 28, 33 | syl2anc 408 | . . . . . . 7 |
35 | 23, 32, 34 | 3imtr4d 202 | . . . . . 6 |
36 | 35 | ex 114 | . . . . 5 |
37 | 36 | com23 78 | . . . 4 |
38 | 17, 37 | mpdd 41 | . . 3 |
39 | 38, 17 | jcad 305 | . 2 |
40 | 5, 39 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 wne 2308 c0 3363 com 4504 (class class class)co 5774 comu 6311 |
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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-oadd 6317 df-omul 6318 |
This theorem is referenced by: nnmword 6414 ltmpig 7147 |
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