xraddge02 - Mathbox for Thierry Arnoux

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Theorem xraddge02 31079

Description: A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)

**Assertion**
Ref	Expression
xraddge02	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +_𝑒 𝐵)))

**Proof of Theorem xraddge02**
Step	Hyp	Ref	Expression
1		xrleid 12885	. . . 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 11022	. . . . 5 ⊢ 0 ∈ ℝ^*
5	3, 4	jctir 521	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ ℝ^ ∧ 0 ∈ ℝ^*))
6		xle2add 12993	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^) ∧ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*)) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵)))
7	5, 6	mpancom 685	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵)))
8	2, 7	mpand 692	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → (𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵)))
9		xaddid1 12975	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
10	9	breq1d 5084	. . 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 5074 (class class class)co 7275 0cc0 10871 ℝ^*cxr 11008 ≤ cle 11010 +_𝑒 cxad 12846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-xadd 12849

This theorem is referenced by: xrge0addge 31080 esummono 32022 esumle 32026 gsumesum 32027 esumlef 32030 measssd 32183 measunl 32184 carsgsigalem 32282

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