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Mirrors > Home > MPE Home > Th. List > xmul01 | Structured version Visualization version GIF version |
Description: Extended real version of mul01 10976. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmul01 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10845 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xmulval 12780 | . . 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 2736 | . . . 4 ⊢ 0 = 0 | |
5 | 4 | olci 866 | . . 3 ⊢ (𝐴 = 0 ∨ 0 = 0) |
6 | 5 | iftruei 4432 | . 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 2787 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ifcif 4425 class class class wbr 5039 (class class class)co 7191 0cc0 10694 · cmul 10699 +∞cpnf 10829 -∞cmnf 10830 ℝ*cxr 10831 < clt 10832 ·e cxmu 12668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-i2m1 10762 ax-rnegex 10765 ax-cnre 10767 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-pnf 10834 df-mnf 10835 df-xr 10836 df-xmul 12671 |
This theorem is referenced by: xmul02 12823 xmulge0 12839 xmulass 12842 xlemul1a 12843 xadddilem 12849 xadddi2 12852 psmetge0 23164 xmetge0 23196 nmoix 23581 hashxpe 30801 xrge0mulc1cn 31559 esumcst 31697 |
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