xraddge02 - Mathbox for Thierry Arnoux

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Theorem xraddge02 33399

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 13287	. . . 4 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
2	1	adantr 486	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → 𝐴 ≤ 𝐴)
3		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → 𝐴 ∈ ℝ^)
4		0xr 11363	. . . . 5 ⊢ 0 ∈ ℝ^*
5	3, 4	jctir 530	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ ℝ^ ∧ 0 ∈ ℝ^*))
6		xle2add 13396	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 0 ∈ ℝ^) ∧ (𝐴 ∈ ℝ^ ∧ 𝐵 ∈ ℝ^*)) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +_e 0) ≤ (𝐴 +_e 𝐵)))
7	5, 6	mpancom 701	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐴 ∧ 0 ≤ 𝐵) → (𝐴 +_e 0) ≤ (𝐴 +_e 𝐵)))
8	2, 7	mpand 708	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → (𝐴 +_e 0) ≤ (𝐴 +_e 𝐵)))
9		xaddrid 13378	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_e 0) = 𝐴)
10	9	breq1d 5112	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 +_e 0) ≤ (𝐴 +_e 𝐵) ↔ 𝐴 ≤ (𝐴 +_e 𝐵)))
11	10	adantr 486	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_e 0) ≤ (𝐴 +_e 𝐵) ↔ 𝐴 ≤ (𝐴 +_e 𝐵)))
12	8, 11	sylibd 242	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ 𝐵 → 𝐴 ≤ (𝐴 +_e 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∈ wcel 2144 class class class wbr 5102 (class class class)co 7424 0cc0 11207 ℝ^*cxr 11349 ≤ cle 11351 +_e cxad 13246

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8005 df-2nd 8006 df-er 8725 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-xadd 13249

This theorem is used by: xrge0addge 33400 esummono 34736 esumle 34740 gsumesum 34741 esumlef 34744 measssd 34898 measunl 34899 carsgsigalem 34997

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