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 | xaddid1 12635 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) |
4 | 3 | eqcomd 2827 | . 2 ⊢ (𝜑 → 𝐴 = (𝐴 +𝑒 0)) |
5 | 0xr 10688 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*) |
7 | 1, 6 | jca 514 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*)) |
8 | iccssxr 12820 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
9 | xadd0ge.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
10 | 8, 9 | sseldi 3965 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | 1, 10 | jca 514 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
12 | 7, 11 | jca 514 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))) |
13 | 1 | xrleidd 12546 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
14 | pnfxr 10695 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ*) |
16 | iccgelb 12794 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
17 | 6, 15, 9, 16 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐵) |
18 | 13, 17 | jca 514 | . . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵)) |
19 | xle2add 12653 | . . 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 5088 | 1 ⊢ (𝜑 → 𝐴 ≤ (𝐴 +𝑒 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 0cc0 10537 +∞cpnf 10672 ℝ*cxr 10674 ≤ cle 10676 +𝑒 cxad 12506 [,]cicc 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-xadd 12509 df-icc 12746 |
This theorem is referenced by: xadd0ge2 41629 sge0xadd 42737 meassle 42765 ovnsubaddlem1 42872 |
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