xmulcl - Metamath Proof Explorer

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Theorem xmulcl 13303

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 13302	. 2 ⊢ ·_e :(ℝ^* × ℝ^)⟶ℝ^
2	1	fovcl 7538	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ·_e 𝐵) ∈ ℝ^)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 ∈ wcel 2143 (class class class)co 7410 ℝ^*cxr 11246 ·_e cxmu 13140

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-addrcl 11165 ax-mulrcl 11167 ax-rnegex 11175 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-po 5569 df-so 5570 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7982 df-2nd 7983 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-xmul 13143

This theorem is used by: xmulpnf1n 13308 xmulge0 13314 xmulasslem3 13316 xmulass 13317 xlemul1a 13318 xlemul1 13320 xltmul1 13322 xadddi 13325 xadddi2 13327 xmulcld 13332 ge0xmulcl 13494 xrsmcmn 21554 xdivpnfrp 33261