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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0addcld | Structured version Visualization version GIF version |
Description: Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.) |
Ref | Expression |
---|---|
xrge0addcld.a | ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) |
xrge0addcld.b | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
xrge0addcld | ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0addcld.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
2 | elxrge0 12835 | . . . . 5 ⊢ (𝐴 ∈ (0[,]+∞) ↔ (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) | |
3 | 1, 2 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴)) |
4 | 3 | simpld 498 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
5 | xrge0addcld.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
6 | elxrge0 12835 | . . . . 5 ⊢ (𝐵 ∈ (0[,]+∞) ↔ (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) | |
7 | 5, 6 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 0 ≤ 𝐵)) |
8 | 7 | simpld 498 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
9 | 4, 8 | xaddcld 12682 | . 2 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ ℝ*) |
10 | 3 | simprd 499 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) |
11 | 7 | simprd 499 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐵) |
12 | xaddge0 12639 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +𝑒 𝐵)) | |
13 | 4, 8, 10, 11, 12 | syl22anc 837 | . 2 ⊢ (𝜑 → 0 ≤ (𝐴 +𝑒 𝐵)) |
14 | elxrge0 12835 | . 2 ⊢ ((𝐴 +𝑒 𝐵) ∈ (0[,]+∞) ↔ ((𝐴 +𝑒 𝐵) ∈ ℝ* ∧ 0 ≤ (𝐴 +𝑒 𝐵))) | |
15 | 9, 13, 14 | sylanbrc 586 | 1 ⊢ (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 0cc0 10526 +∞cpnf 10661 ℝ*cxr 10663 ≤ cle 10665 +𝑒 cxad 12493 [,]cicc 12729 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-xadd 12496 df-icc 12733 |
This theorem is referenced by: omssubadd 31668 |
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