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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 10860 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
3 | possumd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 2, 3 | posdifd 11020 | . 2 ⊢ (𝜑 → (-𝐵 < 𝐴 ↔ 0 < (𝐴 − -𝐵))) |
5 | 3 | recnd 10460 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | 1 | recnd 10460 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 5, 6 | subnegd 10797 | . . 3 ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
8 | 7 | breq2d 4935 | . 2 ⊢ (𝜑 → (0 < (𝐴 − -𝐵) ↔ 0 < (𝐴 + 𝐵))) |
9 | 4, 8 | bitr2d 272 | 1 ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2048 class class class wbr 4923 (class class class)co 6970 ℝcr 10326 0cc0 10327 + caddc 10330 < clt 10466 − cmin 10662 -cneg 10663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5305 df-po 5319 df-so 5320 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-ltxr 10471 df-sub 10664 df-neg 10665 |
This theorem is referenced by: subfzo0 12967 addmodlteq 13122 |
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