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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 9051 | . . 3 | |
2 | 1red 7774 | . . 3 | |
3 | 1, 2 | readdcld 7788 | . 2 |
4 | elznn0nn 9061 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 1 | biantrurd 303 | . . . . 5 |
7 | 6 | orbi2d 779 | . . . 4 |
8 | 5, 7 | mpbird 166 | . . 3 |
9 | peano2nn0 9010 | . . . . 5 | |
10 | 9 | a1i 9 | . . . 4 |
11 | 1 | adantr 274 | . . . . . . . . 9 |
12 | 1red 7774 | . . . . . . . . 9 | |
13 | 11, 12 | readdcld 7788 | . . . . . . . 8 |
14 | 13 | renegcld 8135 | . . . . . . 7 |
15 | 14 | recnd 7787 | . . . . . 6 |
16 | 11 | recnd 7787 | . . . . . . . . . . . 12 |
17 | 1cnd 7775 | . . . . . . . . . . . 12 | |
18 | 16, 17 | negdid 8079 | . . . . . . . . . . 11 |
19 | 18 | oveq1d 5782 | . . . . . . . . . 10 |
20 | 16 | negcld 8053 | . . . . . . . . . . 11 |
21 | neg1cn 8818 | . . . . . . . . . . . 12 | |
22 | 21 | a1i 9 | . . . . . . . . . . 11 |
23 | 20, 22, 17 | addassd 7781 | . . . . . . . . . 10 |
24 | 19, 23 | eqtrd 2170 | . . . . . . . . 9 |
25 | ax-1cn 7706 | . . . . . . . . . . 11 | |
26 | 1pneg1e0 8824 | . . . . . . . . . . 11 | |
27 | 25, 21, 26 | addcomli 7900 | . . . . . . . . . 10 |
28 | 27 | oveq2i 5778 | . . . . . . . . 9 |
29 | 24, 28 | syl6eq 2186 | . . . . . . . 8 |
30 | 20 | addid1d 7904 | . . . . . . . 8 |
31 | 29, 30 | eqtrd 2170 | . . . . . . 7 |
32 | simpr 109 | . . . . . . 7 | |
33 | 31, 32 | eqeltrd 2214 | . . . . . 6 |
34 | elnn0nn 9012 | . . . . . 6 | |
35 | 15, 33, 34 | sylanbrc 413 | . . . . 5 |
36 | 35 | ex 114 | . . . 4 |
37 | 10, 36 | orim12d 775 | . . 3 |
38 | 8, 37 | mpd 13 | . 2 |
39 | elznn0 9062 | . 2 | |
40 | 3, 38, 39 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wcel 1480 (class class class)co 5767 cc 7611 cr 7612 cc0 7613 c1 7614 caddc 7616 cneg 7927 cn 8713 cn0 8970 cz 9047 |
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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sub 7928 df-neg 7929 df-inn 8714 df-n0 8971 df-z 9048 |
This theorem is referenced by: zaddcllempos 9084 peano2zm 9085 zleltp1 9102 btwnnz 9138 peano2uz2 9151 uzind 9155 uzind2 9156 peano2zd 9169 eluzp1m1 9342 eluzp1p1 9344 peano2uz 9371 zltaddlt1le 9782 fzp1disj 9853 elfzp1b 9870 fzneuz 9874 fzp1nel 9877 fzval3 9974 fzossfzop1 9982 rebtwn2zlemstep 10023 flhalf 10068 frec2uzsucd 10167 zesq 10403 hashfzp1 10563 odd2np1lem 11558 odd2np1 11559 mulsucdiv2z 11571 oddp1d2 11576 zob 11577 ltoddhalfle 11579 |
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