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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-addlt0d | Structured version Visualization version GIF version |
Description: The sum of negative numbers is negative. (Contributed by SN, 25-Jan-2025.) |
Ref | Expression |
---|---|
sn-addlt0d.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
sn-addlt0d.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
sn-addlt0d.1 | ⊢ (𝜑 → 𝐴 < 0) |
sn-addlt0d.2 | ⊢ (𝜑 → 𝐵 < 0) |
Ref | Expression |
---|---|
sn-addlt0d | ⊢ (𝜑 → (𝐴 + 𝐵) < 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sn-addlt0d.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | sn-addlt0d.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | 1, 2 | readdcld 11142 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
4 | 0red 11116 | . 2 ⊢ (𝜑 → 0 ∈ ℝ) | |
5 | sn-addlt0d.2 | . . 3 ⊢ (𝜑 → 𝐵 < 0) | |
6 | sn-ltaddneg 40813 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 < 0 ↔ (𝐴 + 𝐵) < 𝐴)) | |
7 | 2, 1, 6 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐵 < 0 ↔ (𝐴 + 𝐵) < 𝐴)) |
8 | 5, 7 | mpbid 231 | . 2 ⊢ (𝜑 → (𝐴 + 𝐵) < 𝐴) |
9 | sn-addlt0d.1 | . 2 ⊢ (𝜑 → 𝐴 < 0) | |
10 | 3, 1, 4, 8, 9 | lttrd 11274 | 1 ⊢ (𝜑 → (𝐴 + 𝐵) < 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7351 ℝcr 11008 0cc0 11009 + caddc 11012 < clt 11147 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-2 12174 df-3 12175 df-resub 40737 |
This theorem is referenced by: sn-nnne0 40819 |
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