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Mirrors > Home > ILE Home > Th. List > renegcl | Unicode version |
Description: Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.) |
Ref | Expression |
---|---|
renegcl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rnegex 7085 | . 2 | |
2 | recn 7106 | . . . . 5 | |
3 | df-neg 7282 | . . . . . . 7 | |
4 | 3 | eqeq1i 2088 | . . . . . 6 |
5 | recn 7106 | . . . . . . 7 | |
6 | 0cn 7111 | . . . . . . . 8 | |
7 | subadd 7311 | . . . . . . . 8 | |
8 | 6, 7 | mp3an1 1255 | . . . . . . 7 |
9 | 5, 8 | sylan 277 | . . . . . 6 |
10 | 4, 9 | syl5bb 190 | . . . . 5 |
11 | 2, 10 | sylan2 280 | . . . 4 |
12 | eleq1a 2150 | . . . . 5 | |
13 | 12 | adantl 271 | . . . 4 |
14 | 11, 13 | sylbird 168 | . . 3 |
15 | 14 | rexlimdva 2477 | . 2 |
16 | 1, 15 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wrex 2349 (class class class)co 5532 cc 6979 cr 6980 cc0 6981 caddc 6984 cmin 7279 cneg 7280 |
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-resscn 7068 ax-1cn 7069 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-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-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 |
This theorem is referenced by: renegcli 7370 resubcl 7372 negreb 7373 renegcld 7484 negf1o 7486 ltnegcon1 7567 ltnegcon2 7568 lenegcon1 7570 lenegcon2 7571 mullt0 7584 recexre 7678 elnnz 8361 btwnz 8466 supinfneg 8683 infsupneg 8684 supminfex 8685 ublbneg 8698 negm 8700 rpnegap 8766 xnegcl 8899 xnegneg 8900 xltnegi 8902 iooneg 9010 iccneg 9011 icoshftf1o 9013 crim 9745 absnid 9959 absdiflt 9978 absdifle 9979 dfabsmax 10103 max0addsup 10105 negfi 10110 minmax 10112 min1inf 10113 min2inf 10114 infssuzex 10345 |
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