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Mirrors > Home > MPE Home > Th. List > nnarcl | Structured version Visualization version Unicode version |
Description: Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
nnarcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oaword1 7632 | . . . 4 | |
2 | eloni 5733 | . . . . . . 7 | |
3 | ordom 7074 | . . . . . . 7 | |
4 | ordtr2 5768 | . . . . . . 7 | |
5 | 2, 3, 4 | sylancl 694 | . . . . . 6 |
6 | 5 | expd 452 | . . . . 5 |
7 | 6 | adantr 481 | . . . 4 |
8 | 1, 7 | mpd 15 | . . 3 |
9 | oaword2 7633 | . . . . 5 | |
10 | 9 | ancoms 469 | . . . 4 |
11 | eloni 5733 | . . . . . . 7 | |
12 | ordtr2 5768 | . . . . . . 7 | |
13 | 11, 3, 12 | sylancl 694 | . . . . . 6 |
14 | 13 | expd 452 | . . . . 5 |
15 | 14 | adantl 482 | . . . 4 |
16 | 10, 15 | mpd 15 | . . 3 |
17 | 8, 16 | jcad 555 | . 2 |
18 | nnacl 7691 | . 2 | |
19 | 17, 18 | impbid1 215 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wcel 1990 wss 3574 word 5722 con0 5723 (class class class)co 6650 com 7065 coa 7557 |
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-rep 4771 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-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 |
This theorem is referenced by: finxpreclem4 33231 |
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