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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 13276 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrge0subcld.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
3 | 1, 2 | sselid 3941 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
4 | xrge0subcld.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
5 | 1, 4 | sselid 3941 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
6 | 5 | xnegcld 13148 | . . . 4 ⊢ (𝜑 → -𝑒𝐵 ∈ ℝ*) |
7 | 3, 6 | xaddcld 13149 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) |
8 | xrge0subcld.c | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
9 | xsubge0 13109 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) | |
10 | 3, 5, 9 | syl2anc 585 | . . . 4 ⊢ (𝜑 → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
11 | 8, 10 | mpbird 257 | . . 3 ⊢ (𝜑 → 0 ≤ (𝐴 +𝑒 -𝑒𝐵)) |
12 | 7, 11 | jca 513 | . 2 ⊢ (𝜑 → ((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) |
13 | elxrge0 13303 | . 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 397 ∈ wcel 2107 class class class wbr 5104 (class class class)co 7350 0cc0 10985 +∞cpnf 11120 ℝ*cxr 11122 ≤ cle 11124 -𝑒cxne 12959 +𝑒 cxad 12960 [,]cicc 13196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 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 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-1st 7912 df-2nd 7913 df-er 8582 df-en 8818 df-dom 8819 df-sdom 8820 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-xneg 12962 df-xadd 12963 df-icc 13200 |
This theorem is referenced by: carsgclctunlem2 32680 |
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