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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 14076 | . 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 1541 ∈ wcel 2106 (class class class)co 7405 ℂcc 11104 · cmul 11111 2c2 12263 ↑cexp 14023 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-seq 13963 df-exp 14024 |
This theorem is referenced by: sqrecii 14143 sqdivi 14145 sqge0i 14148 lt2sqi 14149 le2sqi 14150 sq11i 14151 sq2 14157 sq3 14158 sq4e2t8 14159 i2 14162 expnass 14168 binom2i 14172 sq10 14220 3dec 14222 nn0le2msqi 14223 nn0opthlem1 14224 nn0opth2i 14227 faclbnd4lem1 14249 sqrtmsq2i 15330 pythagtriplem12 16755 pythagtriplem14 16757 prmlem1 17037 prmlem2 17049 4001prm 17074 mcubic 26341 dquartlem1 26345 quart1lem 26349 quart1 26350 log2ublem3 26442 birthday 26448 bposlem7 26782 bposlem8 26783 bposlem9 26784 ax5seglem7 28182 normlem1 30350 nmopcoadji 31341 dpmul4 32067 hgt750lem2 33652 quad3 34643 cntotbnd 36652 3lexlogpow5ineq1 40907 3lexlogpow5ineq5 40913 flt4lem5e 41394 resqrtvalex 42381 imsqrtvalex 42382 fmtno5lem4 46210 flsqrt5 46248 lighneallem4a 46262 |
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