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Theorem sqrtdiv 11786
Description: Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.)
Assertion
Ref Expression
sqrtdiv |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
Proof of Theorem sqrtdiv
StepHypRef Expression
1 rerpdivcl 10064 . . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
21adantlr 481 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( A / B ) e. RR )
3 elrp 10035 . . . . . 6 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
4 divge0 9193 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 <_ ( A / B ) )
53, 4sylan2b 287 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> 0 <_ ( A / B ) )
6 resqrtcl 11773 . . . . 5 |- ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) -> ( sqr ` ( A / B ) ) e. RR )
72, 5, 6syl2anc 415 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) e. RR )
87recnd 8344 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) e. CC )
9 rpsqrtcl 11785 . . . . 5 |- ( B e. RR+ -> ( sqr ` B ) e. RR+ )
109adantl 277 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) e. RR+ )
1110rpcnd 10078 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) e. CC )
1210rpap0d 10082 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) # 0 )
138, 11, 12divcanap4d 9116 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) / ( sqr ` B ) ) = ( sqr ` ( A / B ) ) )
14 rprege0 10048 . . . . . 6 |- ( B e. RR+ -> ( B e. RR /\ 0 <_ B ) )
1514adantl 277 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( B e. RR /\ 0 <_ B ) )
16 sqrtmul 11779 . . . . 5 |- ( ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) /\ ( B e. RR /\ 0 <_ B ) ) -> ( sqr ` ( ( A / B ) x. B ) ) = ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) )
172, 5, 15, 16syl21anc 1277 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( ( A / B ) x. B ) ) = ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) )
18 simpll 531 . . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> A e. RR )
1918recnd 8344 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> A e. CC )
20 rpcn 10042 . . . . . . 7 |- ( B e. RR+ -> B e. CC )
2120adantl 277 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> B e. CC )
22 rpap0 10050 . . . . . . 7 |- ( B e. RR+ -> B # 0 )
2322adantl 277 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> B # 0 )
2419, 21, 23divcanap1d 9111 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( A / B ) x. B ) = A )
2524fveq2d 5694 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( ( A / B ) x. B ) ) = ( sqr ` A ) )
2617, 25eqtr3d 2273 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) = ( sqr ` A ) )
2726oveq1d 6090 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) / ( sqr ` B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
2813, 27eqtr3d 2273 1 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   x. cmul 8174   < clt 8350   <_ cle 8351   # cap 8899   / cdiv 8992   RR+crp 10033   sqrcsqrt 11740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-rp 10034  df-seqfrec 10863  df-exp 10954  df-rsqrt 11742
This theorem is referenced by:  sqrtdivd  11912
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