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Mirrors > Home > MPE Home > Th. List > x2times | Structured version Visualization version GIF version |
Description: Extended real version of 2times 12391. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
x2times | ⊢ (𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 12318 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1re 11252 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rexadd 13256 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 +𝑒 1) = (1 + 1)) | |
4 | 2, 2, 3 | mp2an 690 | . . . 4 ⊢ (1 +𝑒 1) = (1 + 1) |
5 | 1, 4 | eqtr4i 2757 | . . 3 ⊢ 2 = (1 +𝑒 1) |
6 | 5 | oveq1i 7423 | . 2 ⊢ (2 ·e 𝐴) = ((1 +𝑒 1) ·e 𝐴) |
7 | 1xr 11311 | . . . . 5 ⊢ 1 ∈ ℝ* | |
8 | 0le1 11775 | . . . . 5 ⊢ 0 ≤ 1 | |
9 | 7, 8 | pm3.2i 469 | . . . 4 ⊢ (1 ∈ ℝ* ∧ 0 ≤ 1) |
10 | xadddi2r 13322 | . . . 4 ⊢ (((1 ∈ ℝ* ∧ 0 ≤ 1) ∧ (1 ∈ ℝ* ∧ 0 ≤ 1) ∧ 𝐴 ∈ ℝ*) → ((1 +𝑒 1) ·e 𝐴) = ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴))) | |
11 | 9, 9, 10 | mp3an12 1448 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴))) |
12 | xmullid 13304 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴) | |
13 | 12, 12 | oveq12d 7431 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴)) = (𝐴 +𝑒 𝐴)) |
14 | 11, 13 | eqtrd 2766 | . 2 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
15 | 6, 14 | eqtrid 2778 | 1 ⊢ (𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5143 (class class class)co 7413 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 ℝ*cxr 11285 ≤ cle 11287 2c2 12310 +𝑒 cxad 13135 ·e cxmu 13136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7992 df-2nd 7993 df-er 8723 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-2 12318 df-xneg 13137 df-xadd 13138 df-xmul 13139 |
This theorem is referenced by: psmetge0 24303 xmetge0 24335 metnrmlem3 24862 |
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