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Mirrors > Home > MPE Home > Th. List > x2times | Structured version Visualization version GIF version |
Description: Extended real version of 2times 12400. (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 12327 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1re 11259 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | rexadd 13271 | . . . . 5 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ) → (1 +𝑒 1) = (1 + 1)) | |
4 | 2, 2, 3 | mp2an 692 | . . . 4 ⊢ (1 +𝑒 1) = (1 + 1) |
5 | 1, 4 | eqtr4i 2766 | . . 3 ⊢ 2 = (1 +𝑒 1) |
6 | 5 | oveq1i 7441 | . 2 ⊢ (2 ·e 𝐴) = ((1 +𝑒 1) ·e 𝐴) |
7 | 1xr 11318 | . . . . 5 ⊢ 1 ∈ ℝ* | |
8 | 0le1 11784 | . . . . 5 ⊢ 0 ≤ 1 | |
9 | 7, 8 | pm3.2i 470 | . . . 4 ⊢ (1 ∈ ℝ* ∧ 0 ≤ 1) |
10 | xadddi2r 13337 | . . . 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 1450 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴))) |
12 | xmullid 13319 | . . . 4 ⊢ (𝐴 ∈ ℝ* → (1 ·e 𝐴) = 𝐴) | |
13 | 12, 12 | oveq12d 7449 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴)) = (𝐴 +𝑒 𝐴)) |
14 | 11, 13 | eqtrd 2775 | . 2 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
15 | 6, 14 | eqtrid 2787 | 1 ⊢ (𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℝ*cxr 11292 ≤ cle 11294 2c2 12319 +𝑒 cxad 13150 ·e cxmu 13151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-2 12327 df-xneg 13152 df-xadd 13153 df-xmul 13154 |
This theorem is referenced by: psmetge0 24338 xmetge0 24370 metnrmlem3 24897 |
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