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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 13268 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) |
4 | 3 | eqcomd 2732 | . 2 ⊢ (𝜑 → 𝐴 = (𝐴 +𝑒 0)) |
5 | 0xr 11302 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*) |
7 | 1, 6 | jca 510 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*)) |
8 | iccssxr 13455 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
9 | xadd0ge.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
10 | 8, 9 | sselid 3976 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | 1, 10 | jca 510 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
12 | 7, 11 | jca 510 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))) |
13 | 1 | xrleidd 13179 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
14 | pnfxr 11309 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ*) |
16 | iccgelb 13428 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
17 | 6, 15, 9, 16 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐵) |
18 | 13, 17 | jca 510 | . . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵)) |
19 | xle2add 13286 | . . 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 5167 | 1 ⊢ (𝜑 → 𝐴 ≤ (𝐴 +𝑒 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5145 (class class class)co 7416 0cc0 11149 +∞cpnf 11286 ℝ*cxr 11288 ≤ cle 11290 +𝑒 cxad 13138 [,]cicc 13375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-po 5586 df-so 5587 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-xadd 13141 df-icc 13379 |
This theorem is referenced by: xadd0ge2 44992 sge0xadd 46092 meassle 46120 ovnsubaddlem1 46227 |
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