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| Mirrors > Home > MPE Home > Th. List > x2times | Structured version Visualization version GIF version | ||
| Description: Extended real version of 2times 12402. (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 12329 | . . . 4 ⊢ 2 = (1 + 1) | |
| 2 | 1re 11261 | . . . . 5 ⊢ 1 ∈ ℝ | |
| 3 | rexadd 13274 | . . . . 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 2768 | . . 3 ⊢ 2 = (1 +𝑒 1) |
| 6 | 5 | oveq1i 7441 | . 2 ⊢ (2 ·e 𝐴) = ((1 +𝑒 1) ·e 𝐴) |
| 7 | 1xr 11320 | . . . . 5 ⊢ 1 ∈ ℝ* | |
| 8 | 0le1 11786 | . . . . 5 ⊢ 0 ≤ 1 | |
| 9 | 7, 8 | pm3.2i 470 | . . . 4 ⊢ (1 ∈ ℝ* ∧ 0 ≤ 1) |
| 10 | xadddi2r 13340 | . . . 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 1453 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = ((1 ·e 𝐴) +𝑒 (1 ·e 𝐴))) |
| 12 | xmullid 13322 | . . . 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 2777 | . 2 ⊢ (𝐴 ∈ ℝ* → ((1 +𝑒 1) ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
| 15 | 6, 14 | eqtrid 2789 | 1 ⊢ (𝐴 ∈ ℝ* → (2 ·e 𝐴) = (𝐴 +𝑒 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 ℝ*cxr 11294 ≤ cle 11296 2c2 12321 +𝑒 cxad 13152 ·e cxmu 13153 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-2 12329 df-xneg 13154 df-xadd 13155 df-xmul 13156 |
| This theorem is referenced by: psmetge0 24322 xmetge0 24354 metnrmlem3 24883 |
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