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Theorem sqrtdiv 11602
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 9918 . . . . . 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 9889 . . . . . 6 |- ( B e. RR+ <-> ( B e. RR /\ 0 < B ) )
4 divge0 9052 . . . . . 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 11589 . . . . 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 8207 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( A / B ) ) e. CC )
9 rpsqrtcl 11601 . . . . 5 |- ( B e. RR+ -> ( sqr ` B ) e. RR+ )
109adantl 277 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) e. RR+ )
1110rpcnd 9932 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) e. CC )
1210rpap0d 9936 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` B ) # 0 )
138, 11, 12divcanap4d 8975 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) / ( sqr ` B ) ) = ( sqr ` ( A / B ) ) )
14 rprege0 9902 . . . . . 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 11595 . . . . 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 1272 . . . 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 8207 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> A e. CC )
20 rpcn 9896 . . . . . . 7 |- ( B e. RR+ -> B e. CC )
2120adantl 277 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> B e. CC )
22 rpap0 9904 . . . . . . 7 |- ( B e. RR+ -> B # 0 )
2322adantl 277 . . . . . 6 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> B # 0 )
2419, 21, 23divcanap1d 8970 . . . . 5 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( A / B ) x. B ) = A )
2524fveq2d 5643 . . . 4 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( sqr ` ( ( A / B ) x. B ) ) = ( sqr ` A ) )
2617, 25eqtr3d 2266 . . 3 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) = ( sqr ` A ) )
2726oveq1d 6032 . 2 |- ( ( ( A e. RR /\ 0 <_ A ) /\ B e. RR+ ) -> ( ( ( sqr ` ( A / B ) ) x. ( sqr ` B ) ) / ( sqr ` B ) ) = ( ( sqr ` A ) / ( sqr ` B ) ) )
2813, 27eqtr3d 2266 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 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   x. cmul 8036   < clt 8213   <_ cle 8214   # cap 8760   / cdiv 8851   RR+crp 9887   sqrcsqrt 11556
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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151
This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-rp 9888  df-seqfrec 10709  df-exp 10800  df-rsqrt 11558
This theorem is referenced by:  sqrtdivd  11728
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