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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0subcld | Structured version Visualization version GIF version |
Description: Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.) |
Ref | Expression |
---|---|
xrge0subcld.a | ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) |
xrge0subcld.b | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
xrge0subcld.c | ⊢ (𝜑 → 𝐵 ≤ 𝐴) |
Ref | Expression |
---|---|
xrge0subcld | ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 13275 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrge0subcld.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
3 | 1, 2 | sselid 3940 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
4 | xrge0subcld.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
5 | 1, 4 | sselid 3940 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
6 | 5 | xnegcld 13147 | . . . 4 ⊢ (𝜑 → -𝑒𝐵 ∈ ℝ*) |
7 | 3, 6 | xaddcld 13148 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) |
8 | xrge0subcld.c | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
9 | xsubge0 13108 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) | |
10 | 3, 5, 9 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
11 | 8, 10 | mpbird 256 | . . 3 ⊢ (𝜑 → 0 ≤ (𝐴 +𝑒 -𝑒𝐵)) |
12 | 7, 11 | jca 512 | . 2 ⊢ (𝜑 → ((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) |
13 | elxrge0 13302 | . 2 ⊢ ((𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) | |
14 | 12, 13 | sylibr 233 | 1 ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 class class class wbr 5103 (class class class)co 7349 0cc0 10984 +∞cpnf 11119 ℝ*cxr 11121 ≤ cle 11123 -𝑒cxne 12958 +𝑒 cxad 12959 [,]cicc 13195 |
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 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 |
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-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-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5528 df-po 5542 df-so 5543 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-1st 7911 df-2nd 7912 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-xneg 12961 df-xadd 12962 df-icc 13199 |
This theorem is referenced by: carsgclctunlem2 32692 |
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