xaddcld - Intuitionistic Logic Explorer

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Theorem xaddcld 9670

Description: The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)

**Hypotheses**
Ref	Expression
xaddcld.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xaddcld.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)

**Assertion**
Ref	Expression
xaddcld	⊢ (𝜑 → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)

**Proof of Theorem xaddcld**
Step	Hyp	Ref	Expression
1		xaddcld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xaddcld.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xaddcl 9646	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
4	1, 2, 3	syl2anc 408	1 ⊢ (𝜑 → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 1480 (class class class)co 5774 ℝ^*cxr 7802 +_𝑒 cxad 9560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1re 7717 ax-addrcl 7720 ax-rnegex 7732

This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-pnf 7805 df-mnf 7806 df-xr 7807 df-xadd 9563

This theorem is referenced by: xadd4d 9671 xleaddadd 9673 xrmaxaddlem 11032 xrmaxadd 11033 xrminadd 11047 xrbdtri 11048 bldisj 12573 xblss2ps 12576 xblss2 12577 comet 12671 bdxmet 12673 xmetxp 12679