xaddcld - Intuitionistic Logic Explorer

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Theorem xaddcld 9976

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 9952	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2167 (class class class)co 5925 ℝ^*cxr 8077 +_𝑒 cxad 9862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7987 ax-resscn 7988 ax-1re 7990 ax-addrcl 7993 ax-rnegex 8005

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-pnf 8080 df-mnf 8081 df-xr 8082 df-xadd 9865

This theorem is referenced by: xadd4d 9977 xleaddadd 9979 xrmaxaddlem 11442 xrmaxadd 11443 xrminadd 11457 xrbdtri 11458 bldisj 14721 xblss2ps 14724 xblss2 14725 comet 14819 bdxmet 14821 xmetxp 14827