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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 4089 is used to convert hypothesis of the inference (deduction) form of this theorem, renegcli 10221, 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 10152 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℝ, 𝐴, 1) → -𝐴 = -if(𝐴 ∈ ℝ, 𝐴, 1)) | |
2 | 1 | eleq1d 2672 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℝ, 𝐴, 1) → (-𝐴 ∈ ℝ ↔ -if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ)) |
3 | 1re 9918 | . . . 4 ⊢ 1 ∈ ℝ | |
4 | 3 | elimel 4100 | . . 3 ⊢ if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ |
5 | 4 | renegcli 10221 | . 2 ⊢ -if(𝐴 ∈ ℝ, 𝐴, 1) ∈ ℝ |
6 | 2, 5 | dedth 4089 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ifcif 4036 ℝcr 9814 1c1 9816 -cneg 10146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: resubcl 10224 negreb 10225 renegcld 10336 negn0 10338 negf1o 10339 ltnegcon1 10408 ltnegcon2 10409 lenegcon1 10411 lenegcon2 10412 mullt0 10426 mulge0b 10772 mulle0b 10773 negfi 10850 fiminre 10851 infm3lem 10860 infm3 10861 riotaneg 10879 elnnz 11264 btwnz 11355 ublbneg 11649 supminf 11651 uzwo3 11659 zmax 11661 rebtwnz 11663 rpneg 11739 negelrp 11740 max0sub 11901 xnegcl 11918 xnegneg 11919 xltnegi 11921 rexsub 11938 xnegid 11943 xnegdi 11950 xpncan 11953 xnpcan 11954 xadddi 11997 iooneg 12163 iccneg 12164 icoshftf1o 12166 dfceil2 12502 ceicl 12504 ceige 12506 ceim1l 12508 negmod0 12539 negmod 12577 addmodlteq 12607 crim 13703 cnpart 13828 sqrtneglem 13855 absnid 13886 max0add 13898 absdiflt 13905 absdifle 13906 sqreulem 13947 resinhcl 14725 rpcoshcl 14726 tanhlt1 14729 tanhbnd 14730 remulg 19772 resubdrg 19773 cnheiborlem 22561 evth2 22567 ismbf3d 23227 mbfinf 23238 itgconst 23391 reeff1o 24005 atanbnd 24453 sgnneg 29929 ltflcei 32567 cos2h 32570 iblabsnclem 32643 ftc1anclem1 32655 areacirclem2 32671 areacirclem3 32672 areacirc 32675 mulltgt0 38204 limsupre 38708 stoweidlem10 38903 etransclem46 39173 |
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