xaddcld - Intuitionistic Logic Explorer

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Theorem xaddcld 10269

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 10245	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 𝐵) ∈ ℝ^)
4	1, 2, 3	syl2anc 415	1 ⊢ (𝜑 → (𝐴 +_𝑒 𝐵) ∈ ℝ^*)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2209 (class class class)co 6079 ℝ^*cxr 8353 +_𝑒 cxad 10155

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4247 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1re 8267 ax-addrcl 8270 ax-rnegex 8282

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3714 df-pr 3715 df-op 3717 df-uni 3934 df-iun 4012 df-br 4129 df-opab 4191 df-mpt 4192 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-rn 4783 df-res 4784 df-ima 4785 df-iota 5335 df-fun 5377 df-fn 5378 df-f 5379 df-fv 5383 df-ov 6082 df-oprab 6083 df-mpo 6084 df-1st 6368 df-2nd 6369 df-pnf 8356 df-mnf 8357 df-xr 8358 df-xadd 10158

This theorem is referenced by: xadd4d 10270 xleaddadd 10272 xrmaxaddlem 12009 xrmaxadd 12010 xrminadd 12024 xrbdtri 12025 bldisj 15485 xblss2ps 15488 xblss2 15489 comet 15583 bdxmet 15585 xmetxp 15591 vtxdgfifival 16515