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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 13171 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | xrge0subcld.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
3 | 1, 2 | sselid 3920 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
4 | xrge0subcld.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
5 | 1, 4 | sselid 3920 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
6 | 5 | xnegcld 13043 | . . . 4 ⊢ (𝜑 → -𝑒𝐵 ∈ ℝ*) |
7 | 3, 6 | xaddcld 13044 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) |
8 | xrge0subcld.c | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐴) | |
9 | xsubge0 13004 | . . . . 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 13198 | . 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 2107 class class class wbr 5075 (class class class)co 7284 0cc0 10880 +∞cpnf 11015 ℝ*cxr 11017 ≤ cle 11019 -𝑒cxne 12854 +𝑒 cxad 12855 [,]cicc 13091 |
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 2710 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5490 df-po 5504 df-so 5505 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-1st 7840 df-2nd 7841 df-er 8507 df-en 8743 df-dom 8744 df-sdom 8745 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-xneg 12857 df-xadd 12858 df-icc 13095 |
This theorem is referenced by: carsgclctunlem2 32295 |
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