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Mirrors > Home > ILE Home > Th. List > possumd | 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 8299 | . . 3 ⊢ (𝜑 → -𝐵 ∈ ℝ) |
3 | possumd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | 2, 3 | posdifd 8451 | . 2 ⊢ (𝜑 → (-𝐵 < 𝐴 ↔ 0 < (𝐴 − -𝐵))) |
5 | 3 | recnd 7948 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
6 | 1 | recnd 7948 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
7 | 5, 6 | subnegd 8237 | . . 3 ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
8 | 7 | breq2d 4001 | . 2 ⊢ (𝜑 → (0 < (𝐴 − -𝐵) ↔ 0 < (𝐴 + 𝐵))) |
9 | 4, 8 | bitr2d 188 | 1 ⊢ (𝜑 → (0 < (𝐴 + 𝐵) ↔ -𝐵 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 2141 class class class wbr 3989 (class class class)co 5853 ℝcr 7773 0cc0 7774 + caddc 7777 < clt 7954 − cmin 8090 -cneg 8091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-ltxr 7959 df-sub 8092 df-neg 8093 |
This theorem is referenced by: subfzo0 10198 addmodlteq 10354 |
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