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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 | xaddid1 12622 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → (𝐴 +𝑒 0) = 𝐴) |
4 | 3 | eqcomd 2804 | . 2 ⊢ (𝜑 → 𝐴 = (𝐴 +𝑒 0)) |
5 | 0xr 10677 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → 0 ∈ ℝ*) |
7 | 1, 6 | jca 515 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*)) |
8 | iccssxr 12808 | . . . . . 6 ⊢ (0[,]+∞) ⊆ ℝ* | |
9 | xadd0ge.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) | |
10 | 8, 9 | sseldi 3913 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ*) |
11 | 1, 10 | jca 515 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
12 | 7, 11 | jca 515 | . . 3 ⊢ (𝜑 → ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) ∧ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*))) |
13 | 1 | xrleidd 12533 | . . . 4 ⊢ (𝜑 → 𝐴 ≤ 𝐴) |
14 | pnfxr 10684 | . . . . . 6 ⊢ +∞ ∈ ℝ* | |
15 | 14 | a1i 11 | . . . . 5 ⊢ (𝜑 → +∞ ∈ ℝ*) |
16 | iccgelb 12781 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐵 ∈ (0[,]+∞)) → 0 ≤ 𝐵) | |
17 | 6, 15, 9, 16 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐵) |
18 | 13, 17 | jca 515 | . . 3 ⊢ (𝜑 → (𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵)) |
19 | xle2add 12640 | . . 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 5052 | 1 ⊢ (𝜑 → 𝐴 ≤ (𝐴 +𝑒 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ 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: xadd0ge2 41973 sge0xadd 43074 meassle 43102 ovnsubaddlem1 43209 |
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