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Mirrors > Home > ILE Home > Th. List > peano2z | Unicode version |
Description: Second Peano postulate generalized to integers. (Contributed by NM, 13-Feb-2005.) |
Ref | Expression |
---|---|
peano2z |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 9016 | . . 3 | |
2 | 1red 7749 | . . 3 | |
3 | 1, 2 | readdcld 7763 | . 2 |
4 | elznn0nn 9026 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 1 | biantrurd 303 | . . . . 5 |
7 | 6 | orbi2d 764 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | peano2nn0 8975 | . . . . 5 | |
10 | 9 | a1i 9 | . . . 4 |
11 | 1 | adantr 274 | . . . . . . . . 9 |
12 | 1red 7749 | . . . . . . . . 9 | |
13 | 11, 12 | readdcld 7763 | . . . . . . . 8 |
14 | 13 | renegcld 8110 | . . . . . . 7 |
15 | 14 | recnd 7762 | . . . . . 6 |
16 | 11 | recnd 7762 | . . . . . . . . . . . 12 |
17 | 1cnd 7750 | . . . . . . . . . . . 12 | |
18 | 16, 17 | negdid 8054 | . . . . . . . . . . 11 |
19 | 18 | oveq1d 5757 | . . . . . . . . . 10 |
20 | 16 | negcld 8028 | . . . . . . . . . . 11 |
21 | neg1cn 8789 | . . . . . . . . . . . 12 | |
22 | 21 | a1i 9 | . . . . . . . . . . 11 |
23 | 20, 22, 17 | addassd 7756 | . . . . . . . . . 10 |
24 | 19, 23 | eqtrd 2150 | . . . . . . . . 9 |
25 | ax-1cn 7681 | . . . . . . . . . . 11 | |
26 | 1pneg1e0 8795 | . . . . . . . . . . 11 | |
27 | 25, 21, 26 | addcomli 7875 | . . . . . . . . . 10 |
28 | 27 | oveq2i 5753 | . . . . . . . . 9 |
29 | 24, 28 | syl6eq 2166 | . . . . . . . 8 |
30 | 20 | addid1d 7879 | . . . . . . . 8 |
31 | 29, 30 | eqtrd 2150 | . . . . . . 7 |
32 | simpr 109 | . . . . . . 7 | |
33 | 31, 32 | eqeltrd 2194 | . . . . . 6 |
34 | elnn0nn 8977 | . . . . . 6 | |
35 | 15, 33, 34 | sylanbrc 413 | . . . . 5 |
36 | 35 | ex 114 | . . . 4 |
37 | 10, 36 | orim12d 760 | . . 3 |
38 | 8, 37 | mpd 13 | . 2 |
39 | elznn0 9027 | . 2 | |
40 | 3, 38, 39 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 wcel 1465 (class class class)co 5742 cc 7586 cr 7587 cc0 7588 c1 7589 caddc 7591 cneg 7902 cn 8684 cn0 8935 cz 9012 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 df-inn 8685 df-n0 8936 df-z 9013 |
This theorem is referenced by: zaddcllempos 9049 peano2zm 9050 zleltp1 9067 btwnnz 9103 peano2uz2 9116 uzind 9120 uzind2 9121 peano2zd 9134 eluzp1m1 9305 eluzp1p1 9307 peano2uz 9334 zltaddlt1le 9744 fzp1disj 9815 elfzp1b 9832 fzneuz 9836 fzp1nel 9839 fzval3 9936 fzossfzop1 9944 rebtwn2zlemstep 9985 flhalf 10030 frec2uzsucd 10129 zesq 10365 hashfzp1 10525 odd2np1lem 11481 odd2np1 11482 mulsucdiv2z 11494 oddp1d2 11499 zob 11500 ltoddhalfle 11502 |
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