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Mirrors > Home > MPE Home > Th. List > sqmuld | Structured version Visualization version GIF version |
Description: Distribution of square over multiplication. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
expcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
mulexpd.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
sqmuld | ⊢ (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | mulexpd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | sqmul 13335 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → ((𝐴 · 𝐵)↑2) = ((𝐴↑2) · (𝐵↑2))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 (class class class)co 7016 ℂcc 10381 · cmul 10388 2c2 11540 ↑cexp 13279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-n0 11746 df-z 11830 df-uz 12094 df-seq 13220 df-exp 13280 |
This theorem is referenced by: sqoddm1div8 13454 sqrtmul 14453 sqreulem 14553 bhmafibid1cn 14657 bhmafibid2cn 14658 bhmafibid1 14659 pythagtriplem1 15982 prmreclem1 16081 ipcau2 23520 csbren 23685 chordthmlem4 25094 heron 25097 quad2 25098 dquart 25112 cxp2limlem 25235 basellem8 25347 lgsdir 25590 2sqlem3 25678 2sqlem4 25679 2sqlem8 25684 2sqblem 25689 2sqmod 25694 axsegconlem9 26394 ax5seglem1 26397 ax5seglem2 26398 ax5seglem3 26400 rrndstprj2 34660 pellexlem6 38935 pell1234qrne0 38954 pell1234qrreccl 38955 pell1234qrmulcl 38956 pell14qrgt0 38960 pell14qrdich 38970 rmxyneg 39021 wallispi2lem1 41918 stirlinglem3 41923 stirlinglem10 41930 itscnhlc0yqe 44247 itschlc0yqe 44248 itsclc0yqsollem1 44250 itsclc0xyqsolr 44257 itsclquadb 44264 2itscplem1 44266 2itscplem2 44267 itscnhlinecirc02plem1 44270 |
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