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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 4584 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 11570, 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 11500 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℝ, 𝐴, 1) → -𝐴 = -if(𝐴 ∈ ℝ, 𝐴, 1)) | |
| 2 | 1 | eleq1d 2826 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℝ, 𝐴, 1) → (-𝐴 ∈ ℝ ↔ -if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ)) |
| 3 | 1re 11261 | . . . 4 ⊢ 1 ∈ ℝ | |
| 4 | 3 | elimel 4595 | . . 3 ⊢ if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ |
| 5 | 4 | renegcli 11570 | . 2 ⊢ -if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ |
| 6 | 2, 5 | dedth 4584 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ifcif 4525 ℝcr 11154 1c1 11156 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: resubcl 11573 negreb 11574 renegcld 11690 negn0 11692 negf1o 11693 ltnegcon1 11764 ltnegcon2 11765 lenegcon1 11767 lenegcon2 11768 mullt0 11782 mulge0b 12138 mulle0b 12139 negfi 12217 infm3lem 12226 infm3 12227 riotaneg 12247 elnnz 12623 btwnz 12721 ublbneg 12975 supminf 12977 uzwo3 12985 zmax 12987 rebtwnz 12989 rpneg 13067 negelrp 13068 max0sub 13238 xnegcl 13255 xnegneg 13256 xltnegi 13258 rexsub 13275 xnegid 13280 xnegdi 13290 xpncan 13293 xnpcan 13294 xadddi 13337 iooneg 13511 iccneg 13512 icoshftf1o 13514 dfceil2 13879 ceicl 13881 ceige 13884 ceim1l 13887 negmod0 13918 negmod 13957 addmodlteq 13987 crim 15154 cnpart 15279 sqrtneglem 15305 absnid 15337 max0add 15349 absdiflt 15356 absdifle 15357 sqreulem 15398 resinhcl 16192 rpcoshcl 16193 tanhlt1 16196 tanhbnd 16197 remulg 21625 resubdrg 21626 cnheiborlem 24986 evth2 24992 ismbf3d 25689 mbfinf 25700 itgconst 25854 reeff1o 26491 atanbnd 26969 sgnneg 34543 ltflcei 37615 cos2h 37618 iblabsnclem 37690 ftc1anclem1 37700 areacirclem2 37716 areacirclem3 37717 areacirc 37720 mulltgt0 45027 rexabslelem 45429 xnegrecl 45449 supminfrnmpt 45456 supminfxr 45475 limsupre 45656 climinf3 45731 liminfreuzlem 45817 stoweidlem10 46025 etransclem46 46295 smfinflem 46832 finfdm 46861 ceildivmod 47341 line2 48673 |
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