xraddge02 - Mathbox for Thierry Arnoux

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Theorem xraddge02 32476

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 13136	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
2	1	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ 𝐴)
3		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		0xr 11265	. . . . 5 ⊢ 0 ∈ ℝ^*
5	3, 4	jctir 520	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ ℝ^ ∧ 0 ∈ ℝ^*))
6		xle2add 13244	. . . 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		xaddrid 13226	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)
10	9	breq1d 5151	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 +_𝑒 0) ≤ (𝐴 +_𝑒 𝐵) ↔ 𝐴 ≤ (𝐴 +_𝑒 𝐵)))
11	10	adantr 480	. 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 395 ∈ wcel 2098 class class class wbr 5141 (class class class)co 7405 0cc0 11112 ℝ^*cxr 11251 ≤ cle 11253 +_𝑒 cxad 13096

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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188

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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-xadd 13099

This theorem is referenced by: xrge0addge 32477 esummono 33582 esumle 33586 gsumesum 33587 esumlef 33590 measssd 33743 measunl 33744 carsgsigalem 33844

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