xmulcld - Metamath Proof Explorer

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Theorem xmulcld 13284

Description: Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

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

**Assertion**
Ref	Expression
xmulcld	⊢ (𝜑 → (𝐴 ·_e 𝐵) ∈ ℝ^*)

**Proof of Theorem xmulcld**
Step	Hyp	Ref	Expression
1		xnegcld.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xaddcld.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xmulcl 13255	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → (𝐴 ·_e 𝐵) ∈ ℝ^*)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2098 (class class class)co 7404 ℝ^*cxr 11248 ·_e cxmu 13094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-addrcl 11170 ax-mulrcl 11172 ax-rnegex 11180 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-xmul 13097

This theorem is referenced by: nmoi2 24598 nmoleub2lem 24992 xreceu 32591 xdivrec 32596 xrge0adddir 32694 xrmulc1cn 33440 esumcst 33591