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Mirrors > Home > MPE Home > Th. List > x2times | Structured version Visualization version GIF version |
Description: Extended real version of 2times 11145. (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 11079 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1re 10039 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rexadd 12063 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 +𝑒 1) = (1 + 1)) | |
4 | 2, 2, 3 | mp2an 708 | . . . 4 ⊢ (1 +𝑒 1) = (1 + 1) |
5 | 1, 4 | eqtr4i 2647 | . . 3 ⊢ 2 = (1 +𝑒 1) |
6 | 5 | oveq1i 6660 | . 2 ⊢ (2 ·e 𝐴) = ((1 +𝑒 1) ·e 𝐴) |
7 | 2 | rexri 10097 | . . . . 5 ⊢ 1 ∈ ℝ* |
8 | 0le1 10551 | . . . . 5 ⊢ 0 ≤ 1 | |
9 | 7, 8 | pm3.2i 471 | . . . 4 ⊢ (1 ∈ ℝ* ∧ 0 ≤ 1) |
10 | xadddi2r 12128 | . . . 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 1414 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴))) |
12 | xmulid2 12110 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴) | |
13 | 12, 12 | oveq12d 6668 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴)) = (𝐴 +𝑒 𝐴)) |
14 | 11, 13 | eqtrd 2656 | . 2 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
15 | 6, 14 | syl5eq 2668 | 1 ⊢ (𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ*cxr 10073 ≤ cle 10075 2c2 11070 +𝑒 cxad 11944 ·e cxmu 11945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-2 11079 df-xneg 11946 df-xadd 11947 df-xmul 11948 |
This theorem is referenced by: psmetge0 22117 xmetge0 22149 metnrmlem3 22664 |
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