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Mirrors > Home > MPE Home > Th. List > le0neg1 | Structured version Visualization version GIF version |
Description: Comparison of a number and its negative to zero. (Contributed by NM, 10-May-2004.) |
Ref | Expression |
---|---|
le0neg1 | ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10635 | . . 3 ⊢ 0 ∈ ℝ | |
2 | leneg 11135 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 ≤ 0 ↔ -0 ≤ -𝐴)) | |
3 | 1, 2 | mpan2 689 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ -0 ≤ -𝐴)) |
4 | neg0 10924 | . . 3 ⊢ -0 = 0 | |
5 | 4 | breq1i 5064 | . 2 ⊢ (-0 ≤ -𝐴 ↔ 0 ≤ -𝐴) |
6 | 3, 5 | syl6bb 289 | 1 ⊢ (𝐴 ∈ ℝ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2108 class class class wbr 5057 ℝcr 10528 0cc0 10529 ≤ cle 10668 -cneg 10863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 |
This theorem is referenced by: le0neg1d 11203 mulge0b 11502 mulle0b 11503 zle0orge1 11990 max0sub 12581 absnid 14650 max0add 14662 sqreulem 14711 dvdslelem 15651 divalglem1 15737 sqrtnegnre 43497 |
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