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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 14152 | . 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 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 · cmul 11158 2c2 12319 ↑cexp 14099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-exp 14100 |
This theorem is referenced by: sqrecii 14219 sqdivi 14221 sqge0i 14224 lt2sqi 14225 le2sqi 14226 sq11i 14227 sq2 14233 sq3 14234 sq4e2t8 14235 i2 14238 expnass 14244 binom2i 14248 sq10 14300 3dec 14302 nn0le2msqi 14303 nn0opthlem1 14304 nn0opth2i 14307 faclbnd4lem1 14329 sqrtmsq2i 15423 pythagtriplem12 16860 pythagtriplem14 16862 prmlem1 17142 prmlem2 17154 4001prm 17179 mcubic 26905 dquartlem1 26909 quart1lem 26913 quart1 26914 log2ublem3 27006 birthday 27012 bposlem7 27349 bposlem8 27350 bposlem9 27351 ax5seglem7 28965 normlem1 31139 nmopcoadji 32130 dpmul4 32881 hgt750lem2 34646 quad3 35655 cntotbnd 37783 3lexlogpow5ineq1 42036 3lexlogpow5ineq5 42042 sq4 42306 sq5 42307 sq6 42308 sq7 42309 sq8 42310 sq9 42311 flt4lem5e 42643 sq45 42658 resqrtvalex 43635 imsqrtvalex 43636 fmtno5lem4 47481 flsqrt5 47519 lighneallem4a 47533 |
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