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Theorem sqrtdiv 11568
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 9892 . . . . . 6 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
21adantlr 477 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( A / B ) e. RR )
3 elrp 9863 . . . . . 6 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
4 divge0 9031 . . . . . 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 11555 . . . . 5 |- ( ( ( A / B ) e. RR /\ 0 <_ ( A / B ) ) -> ( sqr ` ( A / B ) ) e. RR )
72, 5, 6syl2anc 411 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) e. RR )
87recnd 8186 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) e. CC )
9 rpsqrtcl 11567 . . . . 5 |- ( B e. RR+ -> ( sqr ` B ) e. RR+ )
109adantl 277 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) e. RR+ )
1110rpcnd 9906 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) e. CC )
1210rpap0d 9910 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) # 0 )
138, 11, 12divcanap4d 8954 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) / ( sqr ` B ) ) = ( sqr ` ( A / B ) ) )
14 rprege0 9876 . . . . . 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 11561 . . . . 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 1270 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( ( A / B ) x. B ) ) = ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) )
18 simpll 527 . . . . . . 7 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> A e. RR )
1918recnd 8186 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> A e. CC )
20 rpcn 9870 . . . . . . 7 |- ( B e. RR+ -> B e. CC )
2120adantl 277 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> B e. CC )
22 rpap0 9878 . . . . . . 7 |- ( B e. RR+ -> B # 0 )
2322adantl 277 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> B # 0 )
2419, 21, 23divcanap1d 8949 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( A / B ) x. B ) = A )
2524fveq2d 5633 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( ( A / B ) x. B ) ) = ( sqr ` A ) )
2617, 25eqtr3d 2264 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) = ( sqr ` A ) )
2726oveq1d 6022 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) / ( sqr ` B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
2813, 27eqtr3d 2264 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 1395   e. wcel 2200   class class class wbr 4083   ` cfv 5318  (class class class)co 6007   CCcc 8008   RRcr 8009   0cc0 8010   x. cmul 8015   < clt 8192   <_ cle 8193   # cap 8739   / cdiv 8830   RR+crp 9861   sqrcsqrt 11522
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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129  ax-caucvg 8130
This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-n0 9381  df-z 9458  df-uz 9734  df-rp 9862  df-seqfrec 10682  df-exp 10773  df-rsqrt 11524
This theorem is referenced by:  sqrtdivd  11694
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