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Mirrors > Home > ILE Home > Th. List > sqrtdiv | GIF version |
Description: Square root distributes over division. (Contributed by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
sqrtdiv | ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rerpdivcl 9227 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) | |
2 | 1 | adantlr 462 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (𝐴 / 𝐵) ∈ ℝ) |
3 | elrp 9199 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ ↔ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) | |
4 | divge0 8397 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → 0 ≤ (𝐴 / 𝐵)) | |
5 | 3, 4 | sylan2b 282 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → 0 ≤ (𝐴 / 𝐵)) |
6 | resqrtcl 10525 | . . . . 5 ⊢ (((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) → (√‘(𝐴 / 𝐵)) ∈ ℝ) | |
7 | 2, 5, 6 | syl2anc 404 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) ∈ ℝ) |
8 | 7 | recnd 7579 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) ∈ ℂ) |
9 | rpsqrtcl 10537 | . . . . 5 ⊢ (𝐵 ∈ ℝ+ → (√‘𝐵) ∈ ℝ+) | |
10 | 9 | adantl 272 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘𝐵) ∈ ℝ+) |
11 | 10 | rpcnd 9238 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘𝐵) ∈ ℂ) |
12 | 10 | rpap0d 9242 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘𝐵) # 0) |
13 | 8, 11, 12 | divcanap4d 8326 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (((√‘(𝐴 / 𝐵)) · (√‘𝐵)) / (√‘𝐵)) = (√‘(𝐴 / 𝐵))) |
14 | rprege0 9211 | . . . . . 6 ⊢ (𝐵 ∈ ℝ+ → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
15 | 14 | adantl 272 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) |
16 | sqrtmul 10531 | . . . . 5 ⊢ ((((𝐴 / 𝐵) ∈ ℝ ∧ 0 ≤ (𝐴 / 𝐵)) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → (√‘((𝐴 / 𝐵) · 𝐵)) = ((√‘(𝐴 / 𝐵)) · (√‘𝐵))) | |
17 | 2, 5, 15, 16 | syl21anc 1174 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘((𝐴 / 𝐵) · 𝐵)) = ((√‘(𝐴 / 𝐵)) · (√‘𝐵))) |
18 | simpll 497 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → 𝐴 ∈ ℝ) | |
19 | 18 | recnd 7579 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → 𝐴 ∈ ℂ) |
20 | rpcn 9205 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ) | |
21 | 20 | adantl 272 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → 𝐵 ∈ ℂ) |
22 | rpap0 9213 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 # 0) | |
23 | 22 | adantl 272 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → 𝐵 # 0) |
24 | 19, 21, 23 | divcanap1d 8321 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
25 | 24 | fveq2d 5324 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘((𝐴 / 𝐵) · 𝐵)) = (√‘𝐴)) |
26 | 17, 25 | eqtr3d 2123 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → ((√‘(𝐴 / 𝐵)) · (√‘𝐵)) = (√‘𝐴)) |
27 | 26 | oveq1d 5683 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (((√‘(𝐴 / 𝐵)) · (√‘𝐵)) / (√‘𝐵)) = ((√‘𝐴) / (√‘𝐵))) |
28 | 13, 27 | eqtr3d 2123 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℝ+) → (√‘(𝐴 / 𝐵)) = ((√‘𝐴) / (√‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1290 ∈ wcel 1439 class class class wbr 3853 ‘cfv 5030 (class class class)co 5668 ℂcc 7411 ℝcr 7412 0cc0 7413 · cmul 7418 < clt 7585 ≤ cle 7586 # cap 8121 / cdiv 8202 ℝ+crp 9197 √csqrt 10492 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3962 ax-sep 3965 ax-nul 3973 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-iinf 4418 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 ax-pre-mulext 7526 ax-arch 7527 ax-caucvg 7528 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2624 df-sbc 2844 df-csb 2937 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-nul 3290 df-if 3400 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-int 3697 df-iun 3740 df-br 3854 df-opab 3908 df-mpt 3909 df-tr 3945 df-id 4131 df-po 4134 df-iso 4135 df-iord 4204 df-on 4206 df-ilim 4207 df-suc 4209 df-iom 4421 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-rn 4465 df-res 4466 df-ima 4467 df-iota 4995 df-fun 5032 df-fn 5033 df-f 5034 df-f1 5035 df-fo 5036 df-f1o 5037 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-1st 5927 df-2nd 5928 df-recs 6086 df-frec 6172 df-pnf 7587 df-mnf 7588 df-xr 7589 df-ltxr 7590 df-le 7591 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 df-div 8203 df-inn 8486 df-2 8544 df-3 8545 df-4 8546 df-n0 8737 df-z 8814 df-uz 9083 df-rp 9198 df-iseq 9916 df-seq3 9917 df-exp 10018 df-rsqrt 10494 |
This theorem is referenced by: sqrtdivd 10664 |
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