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Mirrors > Home > MPE Home > Th. List > renegcl | Structured version Visualization version GIF version |
Description: Closure law for negative of reals. The weak deduction theorem dedth 4549 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 11469, to an antecedent. (Contributed by NM, 20-Jan-1997.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
renegcl | ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negeq 11400 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℝ, 𝐴, 1) → -𝐴 = -if(𝐴 ∈ ℝ, 𝐴, 1)) | |
2 | 1 | eleq1d 2823 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℝ, 𝐴, 1) → (-𝐴 ∈ ℝ ↔ -if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ)) |
3 | 1re 11162 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | 3 | elimel 4560 | . . 3 ⊢ if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ |
5 | 4 | renegcli 11469 | . 2 ⊢ -if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ |
6 | 2, 5 | dedth 4549 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ifcif 4491 ℝcr 11057 1c1 11059 -cneg 11393 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-ltxr 11201 df-sub 11394 df-neg 11395 |
This theorem is referenced by: resubcl 11472 negreb 11473 renegcld 11589 negn0 11591 negf1o 11592 ltnegcon1 11663 ltnegcon2 11664 lenegcon1 11666 lenegcon2 11667 mullt0 11681 mulge0b 12032 mulle0b 12033 negfi 12111 infm3lem 12120 infm3 12121 riotaneg 12141 elnnz 12516 btwnz 12613 ublbneg 12865 supminf 12867 uzwo3 12875 zmax 12877 rebtwnz 12879 rpneg 12954 negelrp 12955 max0sub 13122 xnegcl 13139 xnegneg 13140 xltnegi 13142 rexsub 13159 xnegid 13164 xnegdi 13174 xpncan 13177 xnpcan 13178 xadddi 13221 iooneg 13395 iccneg 13396 icoshftf1o 13398 dfceil2 13751 ceicl 13753 ceige 13756 ceim1l 13759 negmod0 13790 negmod 13828 addmodlteq 13858 crim 15007 cnpart 15132 sqrtneglem 15158 absnid 15190 max0add 15202 absdiflt 15209 absdifle 15210 sqreulem 15251 resinhcl 16045 rpcoshcl 16046 tanhlt1 16049 tanhbnd 16050 remulg 21027 resubdrg 21028 cnheiborlem 24333 evth2 24339 ismbf3d 25034 mbfinf 25045 itgconst 25199 reeff1o 25822 atanbnd 26292 sgnneg 33180 ltflcei 36095 cos2h 36098 iblabsnclem 36170 ftc1anclem1 36180 areacirclem2 36196 areacirclem3 36197 areacirc 36200 mulltgt0 43301 rexabslelem 43727 xnegrecl 43747 supminfrnmpt 43754 supminfxr 43773 limsupre 43956 climinf3 44031 liminfreuzlem 44117 stoweidlem10 44325 etransclem46 44595 smfinflem 45132 finfdm 45161 line2 46912 |
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