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Mirrors > Home > ILE Home > Th. List > renegcli | GIF version |
Description: Closure law for negative of reals. (Note: this inference proof style and the deduction theorem usage in renegcl 8266 is deprecated, but is retained for its demonstration value.) (Contributed by NM, 17-Jan-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
renegcl.1 | ⊢ 𝐴 ∈ ℝ |
Ref | Expression |
---|---|
renegcli | ⊢ -𝐴 ∈ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcl.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | renegcl 8266 | . 2 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ -𝐴 ∈ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2160 ℝcr 7857 -cneg 8177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4143 ax-pow 4199 ax-pr 4234 ax-setind 4561 ax-resscn 7950 ax-1cn 7951 ax-icn 7953 ax-addcl 7954 ax-addrcl 7955 ax-mulcl 7956 ax-addcom 7958 ax-addass 7960 ax-distr 7962 ax-i2m1 7963 ax-0id 7966 ax-rnegex 7967 ax-cnre 7969 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2758 df-sbc 2982 df-dif 3151 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-uni 3832 df-br 4026 df-opab 4087 df-id 4318 df-xp 4657 df-rel 4658 df-cnv 4659 df-co 4660 df-dm 4661 df-iota 5203 df-fun 5244 df-fv 5250 df-riota 5861 df-ov 5909 df-oprab 5910 df-mpo 5911 df-sub 8178 df-neg 8179 |
This theorem is referenced by: resubcli 8268 inelr 8589 cju 8966 neg1rr 9074 sincos2sgn 11883 neghalfpire 14856 coseq0negpitopi 14899 negpitopissre 14918 rpabscxpbnd 15001 ex-fl 15141 |
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