xaddge0 - Metamath Proof Explorer

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Theorem xaddge0 13428

Description: The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)

**Assertion**
Ref	Expression
xaddge0	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_e 𝐵))

**Proof of Theorem xaddge0**
Step	Hyp	Ref	Expression
1		0xr 11395	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ ℝ^)
3		simplr 781	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ^)
4		xaddcl 13409	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_e 𝐵) ∈ ℝ^)
5	4	adantr 486	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (𝐴 +_e 𝐵) ∈ ℝ^)
6		simprr 785	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
7		xaddlid 13412	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (0 +_e 𝐵) = 𝐵)
8	3, 7	syl 18	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_e 𝐵) = 𝐵)
9		simpll 779	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐴 ∈ ℝ^)
10		simprl 783	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐴)
11		xleadd1a 13423	. . . 4 ⊢ (((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐴) → (0 +_e 𝐵) ≤ (𝐴 +_e 𝐵))
12	2, 9, 3, 10, 11	syl31anc 1401	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_e 𝐵) ≤ (𝐴 +_e 𝐵))
13	8, 12	eqbrtrrd 5135	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ≤ (𝐴 +_e 𝐵))
14	2, 3, 5, 6, 13	xrletrd 13331	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_e 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 class class class wbr 5108 (class class class)co 7451 0cc0 11239 ℝ^*cxr 11381 ≤ cle 11383 +_e cxad 13279

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-po 5566 df-so 5567 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-ov 7454 df-oprab 7455 df-mpo 7456 df-1st 8035 df-2nd 8036 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-xadd 13282

This theorem is used by: ge0xaddcl 13633 xrge0addcld 33450

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