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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 6112 | . . . . . 6 | |
2 | 1 | ex 113 | . . . . 5 |
3 | 2 | com23 77 | . . . 4 |
4 | 3 | impd 251 | . . 3 |
5 | 4 | 3adant1 956 | . 2 |
6 | ne0i 3257 | . . . . . . . 8 | |
7 | nnm0r 6081 | . . . . . . . . . 10 | |
8 | oveq1 5539 | . . . . . . . . . . 11 | |
9 | 8 | eqeq1d 2089 | . . . . . . . . . 10 |
10 | 7, 9 | syl5ibrcom 155 | . . . . . . . . 9 |
11 | 10 | necon3d 2289 | . . . . . . . 8 |
12 | 6, 11 | syl5 32 | . . . . . . 7 |
13 | 12 | adantr 270 | . . . . . 6 |
14 | nn0eln0 4359 | . . . . . . 7 | |
15 | 14 | adantl 271 | . . . . . 6 |
16 | 13, 15 | sylibrd 167 | . . . . 5 |
17 | 16 | 3adant1 956 | . . . 4 |
18 | oveq2 5540 | . . . . . . . . . 10 | |
19 | 18 | a1i 9 | . . . . . . . . 9 |
20 | nnmordi 6112 | . . . . . . . . . 10 | |
21 | 20 | 3adantl2 1095 | . . . . . . . . 9 |
22 | 19, 21 | orim12d 732 | . . . . . . . 8 |
23 | 22 | con3d 593 | . . . . . . 7 |
24 | simpl3 943 | . . . . . . . . 9 | |
25 | simpl1 941 | . . . . . . . . 9 | |
26 | nnmcl 6083 | . . . . . . . . 9 | |
27 | 24, 25, 26 | syl2anc 403 | . . . . . . . 8 |
28 | simpl2 942 | . . . . . . . . 9 | |
29 | nnmcl 6083 | . . . . . . . . 9 | |
30 | 24, 28, 29 | syl2anc 403 | . . . . . . . 8 |
31 | nntri2 6096 | . . . . . . . 8 | |
32 | 27, 30, 31 | syl2anc 403 | . . . . . . 7 |
33 | nntri2 6096 | . . . . . . . 8 | |
34 | 25, 28, 33 | syl2anc 403 | . . . . . . 7 |
35 | 23, 32, 34 | 3imtr4d 201 | . . . . . 6 |
36 | 35 | ex 113 | . . . . 5 |
37 | 36 | com23 77 | . . . 4 |
38 | 17, 37 | mpdd 40 | . . 3 |
39 | 38, 17 | jcad 301 | . 2 |
40 | 5, 39 | impbid 127 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 wne 2245 c0 3251 com 4331 (class class class)co 5532 comu 6022 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-oadd 6028 df-omul 6029 |
This theorem is referenced by: nnmword 6114 ltmpig 6529 |
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