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Mirrors > Home > MPE Home > Th. List > 2t0e0 | Structured version Visualization version GIF version |
Description: 2 times 0 equals 0. (Contributed by David A. Wheeler, 8-Dec-2018.) |
Ref | Expression |
---|---|
2t0e0 | ⊢ (2 · 0) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 12228 | . 2 ⊢ 2 ∈ ℂ | |
2 | 1 | mul01i 11345 | 1 ⊢ (2 · 0) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7357 0cc0 11051 · cmul 11056 2c2 12208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-ltxr 11194 df-2 12216 |
This theorem is referenced by: expmulnbnd 14138 iseraltlem2 15567 fsumcube 15943 2mulprm 16569 1259lem5 17007 smndex2dnrinv 18725 ablsimpgfindlem1 19886 htpycc 24343 pco0 24377 pcohtpylem 24382 pcopt2 24386 pcoass 24387 pcorevlem 24389 pilem2 25811 cospi 25829 sin2pi 25832 pythag 26167 bclbnd 26628 bposlem1 26632 bposlem2 26633 lgsquadlem1 26728 lgsquadlem2 26729 log2sumbnd 26892 pntrlog2bndlem4 26928 finsumvtxdg2size 28498 cdj3lem1 31376 wrdt2ind 31807 420lcm8e840 40468 dirkertrigeqlem3 44331 fourierdlem62 44399 2exp340mod341 45915 1odd 46095 ackval2012 46767 2itscp 46857 |
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