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Mirrors > Home > MPE Home > Th. List > nnmord | Structured version Visualization version 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 7711 | . . . . . 6 | |
2 | 1 | ex 450 | . . . . 5 |
3 | 2 | com23 86 | . . . 4 |
4 | 3 | impd 447 | . . 3 |
5 | 4 | 3adant1 1079 | . 2 |
6 | ne0i 3921 | . . . . . . . 8 | |
7 | nnm0r 7690 | . . . . . . . . . 10 | |
8 | oveq1 6657 | . . . . . . . . . . 11 | |
9 | 8 | eqeq1d 2624 | . . . . . . . . . 10 |
10 | 7, 9 | syl5ibrcom 237 | . . . . . . . . 9 |
11 | 10 | necon3d 2815 | . . . . . . . 8 |
12 | 6, 11 | syl5 34 | . . . . . . 7 |
13 | 12 | adantr 481 | . . . . . 6 |
14 | nnord 7073 | . . . . . . . 8 | |
15 | ord0eln0 5779 | . . . . . . . 8 | |
16 | 14, 15 | syl 17 | . . . . . . 7 |
17 | 16 | adantl 482 | . . . . . 6 |
18 | 13, 17 | sylibrd 249 | . . . . 5 |
19 | 18 | 3adant1 1079 | . . . 4 |
20 | oveq2 6658 | . . . . . . . . . 10 | |
21 | 20 | a1i 11 | . . . . . . . . 9 |
22 | nnmordi 7711 | . . . . . . . . . 10 | |
23 | 22 | 3adantl2 1218 | . . . . . . . . 9 |
24 | 21, 23 | orim12d 883 | . . . . . . . 8 |
25 | 24 | con3d 148 | . . . . . . 7 |
26 | simpl3 1066 | . . . . . . . . 9 | |
27 | simpl1 1064 | . . . . . . . . 9 | |
28 | nnmcl 7692 | . . . . . . . . 9 | |
29 | 26, 27, 28 | syl2anc 693 | . . . . . . . 8 |
30 | simpl2 1065 | . . . . . . . . 9 | |
31 | nnmcl 7692 | . . . . . . . . 9 | |
32 | 26, 30, 31 | syl2anc 693 | . . . . . . . 8 |
33 | nnord 7073 | . . . . . . . . 9 | |
34 | nnord 7073 | . . . . . . . . 9 | |
35 | ordtri2 5758 | . . . . . . . . 9 | |
36 | 33, 34, 35 | syl2an 494 | . . . . . . . 8 |
37 | 29, 32, 36 | syl2anc 693 | . . . . . . 7 |
38 | nnord 7073 | . . . . . . . . 9 | |
39 | nnord 7073 | . . . . . . . . 9 | |
40 | ordtri2 5758 | . . . . . . . . 9 | |
41 | 38, 39, 40 | syl2an 494 | . . . . . . . 8 |
42 | 27, 30, 41 | syl2anc 693 | . . . . . . 7 |
43 | 25, 37, 42 | 3imtr4d 283 | . . . . . 6 |
44 | 43 | ex 450 | . . . . 5 |
45 | 44 | com23 86 | . . . 4 |
46 | 19, 45 | mpdd 43 | . . 3 |
47 | 46, 19 | jcad 555 | . 2 |
48 | 5, 47 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 c0 3915 word 5722 (class class class)co 6650 com 7065 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: nnmword 7713 nnneo 7731 ltmpi 9726 |
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