| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > xmul01 | Structured version Visualization version GIF version | ||
| Description: Extended real version of mul01 11440. (Contributed by Mario Carneiro, 20-Aug-2015.) |
| Ref | Expression |
|---|---|
| xmul01 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 0xr 11308 | . . 3 ⊢ 0 ∈ ℝ* | |
| 2 | xmulval 13267 | . . 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 691 | . 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 2737 | . . . 4 ⊢ 0 = 0 | |
| 5 | 4 | olci 867 | . . 3 ⊢ (𝐴 = 0 ∨ 0 = 0) |
| 6 | 5 | iftruei 4532 | . 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 | eqtrdi 2793 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ifcif 4525 class class class wbr 5143 (class class class)co 7431 0cc0 11155 · cmul 11160 +∞cpnf 11292 -∞cmnf 11293 ℝ*cxr 11294 < clt 11295 ·e cxmu 13153 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-i2m1 11223 ax-rnegex 11226 ax-cnre 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pnf 11297 df-mnf 11298 df-xr 11299 df-xmul 13156 |
| This theorem is referenced by: xmul02 13310 xmulge0 13326 xmulass 13329 xlemul1a 13330 xadddilem 13336 xadddi2 13339 psmetge0 24322 xmetge0 24354 nmoix 24750 hashxpe 32811 xrge0mulc1cn 33940 esumcst 34064 |
| Copyright terms: Public domain | W3C validator |