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Theorem xaddge0 13201

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 11183	. . 3 ⊢ 0 ∈ ℝ^*
2	1	a1i 11	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ∈ ℝ^)
3		simplr 769	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ∈ ℝ^)
4		xaddcl 13182	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
5	4	adantr 480	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
6		simprr 773	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐵)
7		xaddlid 13185	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (0 +_𝑒 𝐵) = 𝐵)
8	3, 7	syl 17	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_𝑒 𝐵) = 𝐵)
9		simpll 767	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐴 ∈ ℝ^)
10		simprl 771	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ 𝐴)
11		xleadd1a 13196	. . . 4 ⊢ (((0 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 0 ≤ 𝐴) → (0 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
12	2, 9, 3, 10, 11	syl31anc 1376	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → (0 +_𝑒 𝐵) ≤ (𝐴 +_𝑒 𝐵))
13	8, 12	eqbrtrrd 5110	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 𝐵 ≤ (𝐴 +_𝑒 𝐵))
14	2, 3, 5, 6, 13	xrletrd 13104	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (0 ≤ 𝐴 ∧ 0 ≤ 𝐵)) → 0 ≤ (𝐴 +_𝑒 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7360 0cc0 11029 ℝ^*cxr 11169 ≤ cle 11171 +_𝑒 cxad 13052

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8636 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-xadd 13055

This theorem is referenced by: ge0xaddcl 13406 xrge0addcld 32850

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