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Mirrors > Home > ILE Home > Th. List > renegcl | GIF version |
Description: Closure law for negative of reals. (Contributed by NM, 20-Jan-1997.) |
Ref | Expression |
---|---|
renegcl | ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-rnegex 7853 | . 2 ⊢ (𝐴 ∈ ℝ → ∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0) | |
2 | recn 7877 | . . . . 5 ⊢ (𝑥 ∈ ℝ → 𝑥 ∈ ℂ) | |
3 | df-neg 8063 | . . . . . . 7 ⊢ -𝐴 = (0 − 𝐴) | |
4 | 3 | eqeq1i 2172 | . . . . . 6 ⊢ (-𝐴 = 𝑥 ↔ (0 − 𝐴) = 𝑥) |
5 | recn 7877 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 0cn 7882 | . . . . . . . 8 ⊢ 0 ∈ ℂ | |
7 | subadd 8092 | . . . . . . . 8 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((0 − 𝐴) = 𝑥 ↔ (𝐴 + 𝑥) = 0)) | |
8 | 6, 7 | mp3an1 1313 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((0 − 𝐴) = 𝑥 ↔ (𝐴 + 𝑥) = 0)) |
9 | 5, 8 | sylan 281 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℂ) → ((0 − 𝐴) = 𝑥 ↔ (𝐴 + 𝑥) = 0)) |
10 | 4, 9 | syl5bb 191 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℂ) → (-𝐴 = 𝑥 ↔ (𝐴 + 𝑥) = 0)) |
11 | 2, 10 | sylan2 284 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-𝐴 = 𝑥 ↔ (𝐴 + 𝑥) = 0)) |
12 | eleq1a 2236 | . . . . 5 ⊢ (𝑥 ∈ ℝ → (-𝐴 = 𝑥 → -𝐴 ∈ ℝ)) | |
13 | 12 | adantl 275 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (-𝐴 = 𝑥 → -𝐴 ∈ ℝ)) |
14 | 11, 13 | sylbird 169 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝐴 + 𝑥) = 0 → -𝐴 ∈ ℝ)) |
15 | 14 | rexlimdva 2581 | . 2 ⊢ (𝐴 ∈ ℝ → (∃𝑥 ∈ ℝ (𝐴 + 𝑥) = 0 → -𝐴 ∈ ℝ)) |
16 | 1, 15 | mpd 13 | 1 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 ∃wrex 2443 (class class class)co 5836 ℂcc 7742 ℝcr 7743 0cc0 7744 + caddc 7747 − cmin 8060 -cneg 8061 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 |
This theorem is referenced by: renegcli 8151 resubcl 8153 negreb 8154 renegcld 8269 negf1o 8271 ltnegcon1 8352 ltnegcon2 8353 lenegcon1 8355 lenegcon2 8356 mullt0 8369 recexre 8467 elnnz 9192 btwnz 9301 supinfneg 9524 infsupneg 9525 supminfex 9526 ublbneg 9542 negm 9544 rpnegap 9613 negelrp 9614 xnegcl 9759 xnegneg 9760 xltnegi 9762 rexsub 9780 xnegid 9786 xnegdi 9795 xpncan 9798 xnpcan 9799 xposdif 9809 iooneg 9915 iccneg 9916 icoshftf1o 9918 crim 10786 absnid 11001 absdiflt 11020 absdifle 11021 dfabsmax 11145 max0addsup 11147 negfi 11155 minmax 11157 mincl 11158 min1inf 11159 min2inf 11160 minabs 11163 minclpr 11164 mingeb 11169 xrminrecl 11200 xrminrpcl 11201 infssuzex 11867 |
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