Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > leneg2d | Structured version Visualization version GIF version |
Description: Negative of one side of 'less than or equal to'. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
Ref | Expression |
---|---|
leneg2d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
leneg2d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
leneg2d | ⊢ (𝜑 → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | leneg2d.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | leneg2d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 2 | renegcld 11069 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
4 | 1, 3 | lenegd 11221 | . 2 ⊢ (𝜑 → (𝐴 ≤ -𝐵 ↔ --𝐵 ≤ -𝐴)) |
5 | 2 | recnd 10671 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
6 | 5 | negnegd 10990 | . . 3 ⊢ (𝜑 → --𝐵 = 𝐵) |
7 | 6 | breq1d 5078 | . 2 ⊢ (𝜑 → (--𝐵 ≤ -𝐴 ↔ 𝐵 ≤ -𝐴)) |
8 | 4, 7 | bitrd 281 | 1 ⊢ (𝜑 → (𝐴 ≤ -𝐵 ↔ 𝐵 ≤ -𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 class class class wbr 5068 ℝcr 10538 ≤ cle 10678 -cneg 10873 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 |
This theorem is referenced by: liminfreuzlem 42090 |
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