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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rnegex 7753 |
. 2
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2 | recn 7777 |
. . . . 5
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3 | df-neg 7960 |
. . . . . . 7
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4 | 3 | eqeq1i 2148 |
. . . . . 6
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5 | recn 7777 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
6 | 0cn 7782 |
. . . . . . . 8
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7 | subadd 7989 |
. . . . . . . 8
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8 | 6, 7 | mp3an1 1303 |
. . . . . . 7
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9 | 5, 8 | sylan 281 |
. . . . . 6
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10 | 4, 9 | syl5bb 191 |
. . . . 5
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11 | 2, 10 | sylan2 284 |
. . . 4
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12 | eleq1a 2212 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 12 | adantl 275 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 11, 13 | sylbird 169 |
. . 3
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15 | 14 | rexlimdva 2552 |
. 2
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16 | 1, 15 | mpd 13 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-neg 7960 |
This theorem is referenced by: renegcli 8048 resubcl 8050 negreb 8051 renegcld 8166 negf1o 8168 ltnegcon1 8249 ltnegcon2 8250 lenegcon1 8252 lenegcon2 8253 mullt0 8266 recexre 8364 elnnz 9088 btwnz 9194 supinfneg 9417 infsupneg 9418 supminfex 9419 ublbneg 9432 negm 9434 rpnegap 9503 negelrp 9504 xnegcl 9645 xnegneg 9646 xltnegi 9648 rexsub 9666 xnegid 9672 xnegdi 9681 xpncan 9684 xnpcan 9685 xposdif 9695 iooneg 9801 iccneg 9802 icoshftf1o 9804 crim 10662 absnid 10877 absdiflt 10896 absdifle 10897 dfabsmax 11021 max0addsup 11023 negfi 11031 minmax 11033 mincl 11034 min1inf 11035 min2inf 11036 minabs 11039 minclpr 11040 xrminrecl 11074 xrminrpcl 11075 infssuzex 11678 |
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