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Mirrors > Home > MPE Home > Th. List > possumd | Structured version Visualization version GIF version |
Description: Condition for a positive sum. (Contributed by Scott Fenton, 16-Dec-2017.) |
Ref | Expression |
---|---|
possumd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
possumd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
possumd | ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | possumd.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
2 | 1 | renegcld 11681 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
3 | possumd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 2, 3 | posdifd 11841 | . 2 ⊢ (𝜑 → (-𝐵 < 𝐴 ↔ 0 < (𝐴 − -𝐵))) |
5 | 3 | recnd 11282 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | 1 | recnd 11282 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 5, 6 | subnegd 11618 | . . 3 ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
8 | 7 | breq2d 5157 | . 2 ⊢ (𝜑 → (0 < (𝐴 − -𝐵) ↔ 0 < (𝐴 + 𝐵))) |
9 | 4, 8 | bitr2d 279 | 1 ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2099 class class class wbr 5145 (class class class)co 7415 ℝcr 11147 0cc0 11148 + caddc 11151 < clt 11288 − cmin 11484 -cneg 11485 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-pnf 11290 df-mnf 11291 df-ltxr 11293 df-sub 11486 df-neg 11487 |
This theorem is referenced by: subfzo0 13802 addmodlteq 13959 |
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