xmulcld - Metamath Proof Explorer

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Theorem xmulcld 13358

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 13329	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → (𝐴 ·_e 𝐵) ∈ ℝ^*)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 (class class class)co 7417 ℝ^*cxr 11270 ·_e cxmu 13166

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-addrcl 11189 ax-mulrcl 11191 ax-rnegex 11199 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-po 5567 df-so 5568 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7990 df-2nd 7991 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-xmul 13169

This theorem is used by: nmoi2 24962 nmoleub2lem 25348 xreceu 33375 xdivrec 33380 xrge0adddir 33466 xrmulc1cn 34448 esumcst 34581