xaddcld - Intuitionistic Logic Explorer

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Theorem xaddcld 9953

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 9929	. 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 2164 (class class class)co 5919 ℝ^*cxr 8055 +_𝑒 cxad 9839

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-cnex 7965 ax-resscn 7966 ax-1re 7968 ax-addrcl 7971 ax-rnegex 7983

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-pnf 8058 df-mnf 8059 df-xr 8060 df-xadd 9842

This theorem is referenced by: xadd4d 9954 xleaddadd 9956 xrmaxaddlem 11406 xrmaxadd 11407 xrminadd 11421 xrbdtri 11422 bldisj 14580 xblss2ps 14583 xblss2 14584 comet 14678 bdxmet 14680 xmetxp 14686