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Mirrors > Home > MPE Home > Th. List > xmulcl | Structured version Visualization version GIF version |
Description: Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmulcl | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmulf 13230 | . 2 ⊢ ·e :(ℝ* × ℝ*)⟶ℝ* | |
2 | 1 | fovcl 7517 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ·e 𝐵) ∈ ℝ*) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 (class class class)co 7390 ℝ*cxr 11226 ·e cxmu 13070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-addrcl 11150 ax-mulrcl 11152 ax-rnegex 11160 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-po 5578 df-so 5579 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7954 df-2nd 7955 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-xmul 13073 |
This theorem is referenced by: xmulpnf1n 13236 xmulge0 13242 xmulasslem3 13244 xmulass 13245 xlemul1a 13246 xlemul1 13248 xltmul1 13250 xadddi 13253 xadddi2 13255 xmulcld 13260 ge0xmulcl 13419 xrsmcmn 20897 xdivpnfrp 31965 |
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