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Mirrors > Home > ILE Home > Th. List > resqrtcl | GIF version |
Description: Closure of the square root function. (Contributed by Mario Carneiro, 9-Jul-2013.) |
Ref | Expression |
---|---|
resqrtcl | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqrex 10513 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃𝑦 ∈ ℝ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) | |
2 | simp1l 968 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → 𝐴 ∈ ℝ) | |
3 | sqrtrval 10487 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) | |
4 | 2, 3 | syl 14 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) |
5 | simp3r 973 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → (𝑦↑2) = 𝐴) | |
6 | simp3l 972 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → 0 ≤ 𝑦) | |
7 | simp2 945 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → 𝑦 ∈ ℝ) | |
8 | rersqreu 10515 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) | |
9 | 8 | 3ad2ant1 965 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) |
10 | oveq1 5673 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2)) | |
11 | 10 | eqeq1d 2097 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) = 𝐴 ↔ (𝑦↑2) = 𝐴)) |
12 | breq2 3855 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (0 ≤ 𝑥 ↔ 0 ≤ 𝑦)) | |
13 | 11, 12 | anbi12d 458 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ↔ ((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦))) |
14 | 13 | riota2 5644 | . . . . . . 7 ⊢ ((𝑦 ∈ ℝ ∧ ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) → (((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦) ↔ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = 𝑦)) |
15 | 7, 9, 14 | syl2anc 404 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → (((𝑦↑2) = 𝐴 ∧ 0 ≤ 𝑦) ↔ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = 𝑦)) |
16 | 5, 6, 15 | mpbi2and 890 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = 𝑦) |
17 | 4, 16 | eqtrd 2121 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → (√‘𝐴) = 𝑦) |
18 | 17, 7 | eqeltrd 2165 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝑦 ∈ ℝ ∧ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴)) → (√‘𝐴) ∈ ℝ) |
19 | 18 | rexlimdv3a 2492 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∃𝑦 ∈ ℝ (0 ≤ 𝑦 ∧ (𝑦↑2) = 𝐴) → (√‘𝐴) ∈ ℝ)) |
20 | 1, 19 | mpd 13 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 ∃wrex 2361 ∃!wreu 2362 class class class wbr 3851 ‘cfv 5028 ℩crio 5621 (class class class)co 5666 ℝcr 7403 0cc0 7404 ≤ cle 7577 2c2 8527 ↑cexp 10008 √csqrt 10483 |
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 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-mulrcl 7498 ax-addcom 7499 ax-mulcom 7500 ax-addass 7501 ax-mulass 7502 ax-distr 7503 ax-i2m1 7504 ax-0lt1 7505 ax-1rid 7506 ax-0id 7507 ax-rnegex 7508 ax-precex 7509 ax-cnre 7510 ax-pre-ltirr 7511 ax-pre-ltwlin 7512 ax-pre-lttrn 7513 ax-pre-apti 7514 ax-pre-ltadd 7515 ax-pre-mulgt0 7516 ax-pre-mulext 7517 ax-arch 7518 ax-caucvg 7519 |
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 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-if 3398 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-ilim 4205 df-suc 4207 df-iom 4419 df-xp 4457 df-rel 4458 df-cnv 4459 df-co 4460 df-dm 4461 df-rn 4462 df-res 4463 df-ima 4464 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-frec 6170 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 df-sub 7709 df-neg 7710 df-reap 8106 df-ap 8113 df-div 8194 df-inn 8477 df-2 8535 df-3 8536 df-4 8537 df-n0 8728 df-z 8805 df-uz 9074 df-rp 9189 df-iseq 9907 df-seq3 9908 df-exp 10009 df-rsqrt 10485 |
This theorem is referenced by: rersqrtthlem 10517 remsqsqrt 10519 sqrtgt0 10521 sqrtmul 10522 sqrtle 10523 sqrtlt 10524 sqrt11ap 10525 sqrt11 10526 rpsqrtcl 10528 sqrtdiv 10529 sqrtsq2 10530 abscl 10538 amgm2 10605 sqrtcli 10607 resqrtcld 10650 |
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