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Mirrors > Home > MPE Home > Th. List > times2d | Structured version Visualization version GIF version |
Description: A number times 2. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
2timesd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
times2d | ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2timesd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | times2 11146 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 · 2) = (𝐴 + 𝐴)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 · 2) = (𝐴 + 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 + caddc 9939 · cmul 9941 2c2 11070 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 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-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rrecex 10008 ax-cnre 10009 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-2 11079 |
This theorem is referenced by: div4p1lem1div2 11287 climcndslem1 14581 climcndslem2 14582 sadcaddlem 15179 dvexp3 23741 chordthmlem 24559 chordthmlem2 24560 chordthmlem4 24562 logfaclbnd 24947 rplogsumlem1 25173 nexple 30071 |
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