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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > xadd0ge | Structured version Visualization version GIF version |
Description: A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xadd0ge.a | ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
xadd0ge.b | ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
xadd0ge | ⊢ (𝜑 → 𝐴 ≤ (𝐴 +𝑒 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xadd0ge.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*) | |
2 | xaddrid 13218 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) |
4 | 3 | eqcomd 2730 | . 2 ⊢ (𝜑 → 𝐴 = (𝐴 +𝑒 0)) |
5 | 0xr 11259 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*) |
7 | 1, 6 | jca 511 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*)) |
8 | iccssxr 13405 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
9 | xadd0ge.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
10 | 8, 9 | sselid 3973 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | 1, 10 | jca 511 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
12 | 7, 11 | jca 511 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))) |
13 | 1 | xrleidd 13129 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
14 | pnfxr 11266 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ*) |
16 | iccgelb 13378 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
17 | 6, 15, 9, 16 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐵) |
18 | 13, 17 | jca 511 | . . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵)) |
19 | xle2add 13236 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵))) | |
20 | 12, 18, 19 | sylc 65 | . 2 ⊢ (𝜑 → (𝐴 +𝑒 0) ≤ (𝐴 +𝑒 𝐵)) |
21 | 4, 20 | eqbrtrd 5161 | 1 ⊢ (𝜑 → 𝐴 ≤ (𝐴 +𝑒 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5139 (class class class)co 7402 0cc0 11107 +∞cpnf 11243 ℝ*cxr 11245 ≤ cle 11247 +𝑒 cxad 13088 [,]cicc 13325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-xadd 13091 df-icc 13329 |
This theorem is referenced by: xadd0ge2 44561 sge0xadd 45661 meassle 45689 ovnsubaddlem1 45796 |
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