Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sqvali | Structured version Visualization version GIF version |
Description: Value of square. Inference version. (Contributed by NM, 1-Aug-1999.) |
Ref | Expression |
---|---|
sqval.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
sqvali | ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqval.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | sqval 13687 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 (class class class)co 7213 ℂcc 10727 · cmul 10734 2c2 11885 ↑cexp 13635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-n0 12091 df-z 12177 df-uz 12439 df-seq 13575 df-exp 13636 |
This theorem is referenced by: sqrecii 13752 sqdivi 13754 sqge0i 13757 lt2sqi 13758 le2sqi 13759 sq11i 13760 sq2 13766 sq3 13767 sq4e2t8 13768 i2 13771 expnass 13776 binom2i 13780 sq10 13830 3dec 13832 nn0le2msqi 13833 nn0opthlem1 13834 nn0opth2i 13837 faclbnd4lem1 13859 sqrtmsq2i 14951 pythagtriplem12 16379 pythagtriplem14 16381 prmlem1 16661 prmlem2 16673 4001prm 16698 mcubic 25730 dquartlem1 25734 quart1lem 25738 quart1 25739 log2ublem3 25831 birthday 25837 bposlem7 26171 bposlem8 26172 bposlem9 26173 ax5seglem7 27026 normlem1 29191 nmopcoadji 30182 dpmul4 30908 hgt750lem2 32344 quad3 33341 cntotbnd 35691 3lexlogpow5ineq1 39796 3lexlogpow5ineq5 39802 flt4lem5e 40196 resqrtvalex 40929 imsqrtvalex 40930 fmtno5lem4 44681 flsqrt5 44719 lighneallem4a 44733 |
Copyright terms: Public domain | W3C validator |