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Mirrors > Home > MPE Home > Th. List > xmul01 | Structured version Visualization version GIF version |
Description: Extended real version of mul01 10822. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmul01 | ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 10691 | . . 3 ⊢ 0 ∈ ℝ* | |
2 | xmulval 12621 | . . 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 689 | . 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 2824 | . . . 4 ⊢ 0 = 0 | |
5 | 4 | olci 862 | . . 3 ⊢ (𝐴 = 0 ∨ 0 = 0) |
6 | 5 | iftruei 4477 | . 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 2875 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 ·e 0) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ifcif 4470 class class class wbr 5069 (class class class)co 7159 0cc0 10540 · cmul 10545 +∞cpnf 10675 -∞cmnf 10676 ℝ*cxr 10677 < clt 10678 ·e cxmu 12509 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-i2m1 10608 ax-rnegex 10611 ax-cnre 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-pnf 10680 df-mnf 10681 df-xr 10682 df-xmul 12512 |
This theorem is referenced by: xmul02 12664 xmulge0 12680 xmulass 12683 xlemul1a 12684 xadddilem 12690 xadddi2 12693 psmetge0 22925 xmetge0 22957 nmoix 23341 hashxpe 30532 xrge0mulc1cn 31188 esumcst 31326 |
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