xmulcl - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xmulcl		Structured version Visualization version GIF version

Theorem xmulcl 13396

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 13395	. 2 ⊢ ·_e :(ℝ^* × ℝ^)⟶ℝ^
2	1	fovcl 7546	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 2145 (class class class)co 7418 ℝ^*cxr 11335 ·_e cxmu 13233

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 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-addrcl 11254 ax-mulrcl 11256 ax-rnegex 11264 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 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 7421 df-oprab 7422 df-mpo 7423 df-1st 7999 df-2nd 8000 df-er 8710 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-xmul 13236

This theorem is used by: xmulpnf1n 13401 xmulge0 13407 xmulasslem3 13409 xmulass 13410 xlemul1a 13411 xlemul1 13413 xltmul1 13415 xadddi 13418 xadddi2 13420 xmulcld 13425 ge0xmulcl 13587 xrsmcmn 21694 xdivpnfrp 33492