xaddcld - Intuitionistic Logic Explorer

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Theorem xaddcld 9880

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 9856	. 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 2148 (class class class)co 5872 ℝ^*cxr 7987 +_𝑒 cxad 9766

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 ax-rnegex 7917

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-fv 5223 df-ov 5875 df-oprab 5876 df-mpo 5877 df-1st 6138 df-2nd 6139 df-pnf 7990 df-mnf 7991 df-xr 7992 df-xadd 9769

This theorem is referenced by: xadd4d 9881 xleaddadd 9883 xrmaxaddlem 11261 xrmaxadd 11262 xrminadd 11276 xrbdtri 11277 bldisj 13772 xblss2ps 13775 xblss2 13776 comet 13870 bdxmet 13872 xmetxp 13878