xmulcl - Metamath Proof Explorer

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Theorem xmulcl 13311

Description: Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xmulcl	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)

**Proof of Theorem xmulcl**
Step	Hyp	Ref	Expression
1		xmulf 13310	. 2 ⊢ ·_e :(ℝ^* × ℝ^)⟶ℝ^
2	1	fovcl 7544	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2146 (class class class)co 7416 ℝ^*cxr 11253 ·_e cxmu 13148

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-addrcl 11172 ax-mulrcl 11174 ax-rnegex 11182 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7988 df-2nd 7989 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-xmul 13151

This theorem is used by: xmulpnf1n 13316 xmulge0 13322 xmulasslem3 13324 xmulass 13325 xlemul1a 13326 xlemul1 13328 xltmul1 13330 xadddi 13333 xadddi2 13335 xmulcld 13340 ge0xmulcl 13502 xrsmcmn 21575 xdivpnfrp 33298