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Mirrors > Home > MPE Home > Th. List > Mathboxes > xraddge02 | Structured version Visualization version GIF version |
Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.) |
Ref | Expression |
---|---|
xraddge02 | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +𝑒 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrleid 13062 | . . . 4 ⊢ (𝐴 ∈ ℝ* → 𝐴 ≤ 𝐴) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ≤ 𝐴) |
3 | simpl 483 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → 𝐴 ∈ ℝ*) | |
4 | 0xr 11198 | . . . . 5 ⊢ 0 ∈ ℝ* | |
5 | 3, 4 | jctir 521 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*)) |
6 | xle2add 13170 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵))) | |
7 | 5, 6 | mpancom 686 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵))) |
8 | 2, 7 | mpand 693 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → (𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵))) |
9 | xaddid1 13152 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
10 | 9 | breq1d 5113 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵) ↔ 𝐴 ≤ (𝐴 +𝑒 𝐵))) |
11 | 10 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵) ↔ 𝐴 ≤ (𝐴 +𝑒 𝐵))) |
12 | 8, 11 | sylibd 238 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +𝑒 𝐵))) |
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 7353 0cc0 11047 ℝ*cxr 11184 ≤ cle 11186 +𝑒 cxad 13023 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 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 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 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-xadd 13026 |
This theorem is referenced by: xrge0addge 31545 esummono 32522 esumle 32526 gsumesum 32527 esumlef 32530 measssd 32683 measunl 32684 carsgsigalem 32784 |
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