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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 8355 | . . 3 | |
2 | 1red 7134 | . . 3 | |
3 | 1, 2 | readdcld 7148 | . 2 |
4 | elznn0nn 8365 | . . . . 5 | |
5 | 4 | biimpi 118 | . . . 4 |
6 | 1 | biantrurd 299 | . . . . 5 |
7 | 6 | orbi2d 736 | . . . 4 |
8 | 5, 7 | mpbird 165 | . . 3 |
9 | peano2nn0 8328 | . . . . 5 | |
10 | 9 | a1i 9 | . . . 4 |
11 | 1 | adantr 270 | . . . . . . . . 9 |
12 | 1red 7134 | . . . . . . . . 9 | |
13 | 11, 12 | readdcld 7148 | . . . . . . . 8 |
14 | 13 | renegcld 7484 | . . . . . . 7 |
15 | 14 | recnd 7147 | . . . . . 6 |
16 | 11 | recnd 7147 | . . . . . . . . . . . 12 |
17 | 1cnd 7135 | . . . . . . . . . . . 12 | |
18 | 16, 17 | negdid 7432 | . . . . . . . . . . 11 |
19 | 18 | oveq1d 5547 | . . . . . . . . . 10 |
20 | 16 | negcld 7406 | . . . . . . . . . . 11 |
21 | neg1cn 8144 | . . . . . . . . . . . 12 | |
22 | 21 | a1i 9 | . . . . . . . . . . 11 |
23 | 20, 22, 17 | addassd 7141 | . . . . . . . . . 10 |
24 | 19, 23 | eqtrd 2113 | . . . . . . . . 9 |
25 | ax-1cn 7069 | . . . . . . . . . . 11 | |
26 | 1pneg1e0 8150 | . . . . . . . . . . 11 | |
27 | 25, 21, 26 | addcomli 7253 | . . . . . . . . . 10 |
28 | 27 | oveq2i 5543 | . . . . . . . . 9 |
29 | 24, 28 | syl6eq 2129 | . . . . . . . 8 |
30 | 20 | addid1d 7257 | . . . . . . . 8 |
31 | 29, 30 | eqtrd 2113 | . . . . . . 7 |
32 | simpr 108 | . . . . . . 7 | |
33 | 31, 32 | eqeltrd 2155 | . . . . . 6 |
34 | elnn0nn 8330 | . . . . . 6 | |
35 | 15, 33, 34 | sylanbrc 408 | . . . . 5 |
36 | 35 | ex 113 | . . . 4 |
37 | 10, 36 | orim12d 732 | . . 3 |
38 | 8, 37 | mpd 13 | . 2 |
39 | elznn0 8366 | . 2 | |
40 | 3, 38, 39 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 wcel 1433 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 c1 6982 caddc 6984 cneg 7280 cn 8039 cn0 8288 cz 8351 |
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-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
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-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 df-inn 8040 df-n0 8289 df-z 8352 |
This theorem is referenced by: zaddcllempos 8388 peano2zm 8389 zleltp1 8406 btwnnz 8441 peano2uz2 8454 uzind 8458 uzind2 8459 peano2zd 8472 eluzp1m1 8642 eluzp1p1 8644 peano2uz 8671 zltaddlt1le 9028 fzp1disj 9097 elfzp1b 9114 fzneuz 9118 fzp1nel 9121 fzval3 9213 fzossfzop1 9221 rebtwn2zlemstep 9261 flhalf 9304 frec2uzzd 9402 frec2uzsucd 9403 zesq 9591 odd2np1lem 10271 odd2np1 10272 mulsucdiv2z 10285 oddp1d2 10290 zob 10291 ltoddhalfle 10293 |
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