Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2timesgt | Structured version Visualization version GIF version |
Description: Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
2timesgt | ⊢ (𝐴 ∈ ℝ+ → 𝐴 < (2 · 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpre 12385 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
2 | id 22 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ+) | |
3 | 1, 2 | ltaddrp2d 12453 | . 2 ⊢ (𝐴 ∈ ℝ+ → 𝐴 < (𝐴 + 𝐴)) |
4 | rpcn 12387 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
5 | 2times 11761 | . . . 4 ⊢ (𝐴 ∈ ℂ → (2 · 𝐴) = (𝐴 + 𝐴)) | |
6 | 5 | eqcomd 2824 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + 𝐴) = (2 · 𝐴)) |
7 | 4, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴 + 𝐴) = (2 · 𝐴)) |
8 | 3, 7 | breqtrd 5083 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 < (2 · 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 (class class class)co 7145 ℂcc 10523 + caddc 10528 · cmul 10530 < clt 10663 2c2 11680 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-2 11688 df-rp 12378 |
This theorem is referenced by: limsup10exlem 41929 fourierdlem24 42293 fourierdlem43 42312 fourierdlem44 42313 sqwvfoura 42390 sqwvfourb 42391 fourierswlem 42392 fouriersw 42393 |
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