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Mirrors > Home > MPE Home > Th. List > xmul01 | Structured version Visualization version GIF version |
Description: Extended real version of mul01 10555. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmul01 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10423 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xmulval 12368 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ∈ ℝ*) → (𝐴 ·e 0) = if((𝐴 = 0 ∨ 0 = 0), 0, if((((0 < 0 ∧ 𝐴 = +∞) ∨ (0 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 0 = +∞) ∨ (𝐴 < 0 ∧ 0 = -∞))), +∞, if((((0 < 0 ∧ 𝐴 = -∞) ∨ (0 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 0 = -∞) ∨ (𝐴 < 0 ∧ 0 = +∞))), -∞, (𝐴 · 0))))) | |
3 | 1, 2 | mpan2 681 | . 2 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = if((𝐴 = 0 ∨ 0 = 0), 0, if((((0 < 0 ∧ 𝐴 = +∞) ∨ (0 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 0 = +∞) ∨ (𝐴 < 0 ∧ 0 = -∞))), +∞, if((((0 < 0 ∧ 𝐴 = -∞) ∨ (0 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 0 = -∞) ∨ (𝐴 < 0 ∧ 0 = +∞))), -∞, (𝐴 · 0))))) |
4 | eqid 2777 | . . . 4 ⊢ 0 = 0 | |
5 | 4 | olci 855 | . . 3 ⊢ (𝐴 = 0 ∨ 0 = 0) |
6 | 5 | iftruei 4313 | . 2 ⊢ if((𝐴 = 0 ∨ 0 = 0), 0, if((((0 < 0 ∧ 𝐴 = +∞) ∨ (0 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 0 = +∞) ∨ (𝐴 < 0 ∧ 0 = -∞))), +∞, if((((0 < 0 ∧ 𝐴 = -∞) ∨ (0 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 0 = -∞) ∨ (𝐴 < 0 ∧ 0 = +∞))), -∞, (𝐴 · 0)))) = 0 |
7 | 3, 6 | syl6eq 2829 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2106 ifcif 4306 class class class wbr 4886 (class class class)co 6922 0cc0 10272 · cmul 10277 +∞cpnf 10408 -∞cmnf 10409 ℝ*cxr 10410 < clt 10411 ·e cxmu 12256 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-i2m1 10340 ax-rnegex 10343 ax-cnre 10345 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-pnf 10413 df-mnf 10414 df-xr 10415 df-xmul 12259 |
This theorem is referenced by: xmul02 12410 xmulge0 12426 xmulass 12429 xlemul1a 12430 xadddilem 12436 xadddi2 12439 psmetge0 22525 xmetge0 22557 nmoix 22941 xrge0mulc1cn 30585 esumcst 30723 |
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