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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 12904 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrge0subcld.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
3 | 1, 2 | sseldi 3875 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
4 | xrge0subcld.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
5 | 1, 4 | sseldi 3875 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
6 | 5 | xnegcld 12776 | . . . 4 ⊢ (𝜑 → -𝑒𝐵 ∈ ℝ*) |
7 | 3, 6 | xaddcld 12777 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) |
8 | xrge0subcld.c | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
9 | xsubge0 12737 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) | |
10 | 3, 5, 9 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) |
11 | 8, 10 | mpbird 260 | . . 3 ⊢ (𝜑 → 0 ≤ (𝐴 +𝑒 -𝑒𝐵)) |
12 | 7, 11 | jca 515 | . 2 ⊢ (𝜑 → ((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) |
13 | elxrge0 12931 | . 2 ⊢ ((𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞) ↔ ((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) | |
14 | 12, 13 | sylibr 237 | 1 ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2114 class class class wbr 5030 (class class class)co 7170 0cc0 10615 +∞cpnf 10750 ℝ*cxr 10752 ≤ cle 10754 -𝑒cxne 12587 +𝑒 cxad 12588 [,]cicc 12824 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-1st 7714 df-2nd 7715 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-xneg 12590 df-xadd 12591 df-icc 12828 |
This theorem is referenced by: carsgclctunlem2 31856 |
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