![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > rersqrtthlem | GIF version |
Description: Lemma for resqrtth 10527. (Contributed by Jim Kingdon, 10-Aug-2021.) |
Ref | Expression |
---|---|
rersqrtthlem | ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrtrval 10496 | . . . 4 ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) | |
2 | 1 | eqcomd 2094 | . . 3 ⊢ (𝐴 ∈ ℝ → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = (√‘𝐴)) |
3 | 2 | adantr 271 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = (√‘𝐴)) |
4 | resqrtcl 10525 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (√‘𝐴) ∈ ℝ) | |
5 | rersqreu 10524 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) | |
6 | oveq1 5675 | . . . . . 6 ⊢ (𝑥 = (√‘𝐴) → (𝑥↑2) = ((√‘𝐴)↑2)) | |
7 | 6 | eqeq1d 2097 | . . . . 5 ⊢ (𝑥 = (√‘𝐴) → ((𝑥↑2) = 𝐴 ↔ ((√‘𝐴)↑2) = 𝐴)) |
8 | breq2 3857 | . . . . 5 ⊢ (𝑥 = (√‘𝐴) → (0 ≤ 𝑥 ↔ 0 ≤ (√‘𝐴))) | |
9 | 7, 8 | anbi12d 458 | . . . 4 ⊢ (𝑥 = (√‘𝐴) → (((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥) ↔ (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴)))) |
10 | 9 | riota2 5646 | . . 3 ⊢ (((√‘𝐴) ∈ ℝ ∧ ∃!𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) → ((((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴)) ↔ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = (√‘𝐴))) |
11 | 4, 5, 10 | syl2anc 404 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴)) ↔ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) = (√‘𝐴))) |
12 | 3, 11 | mpbird 166 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (((√‘𝐴)↑2) = 𝐴 ∧ 0 ≤ (√‘𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1290 ∈ wcel 1439 ∃!wreu 2362 class class class wbr 3853 ‘cfv 5030 ℩crio 5623 (class class class)co 5668 ℝcr 7412 0cc0 7413 ≤ cle 7586 2c2 8536 ↑cexp 10017 √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: resqrtth 10527 sqrtge0 10529 |
Copyright terms: Public domain | W3C validator |