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Mirrors > Home > ILE Home > Th. List > sqrtrval | GIF version |
Description: Value of square root function. (Contributed by Jim Kingdon, 23-Aug-2020.) |
Ref | Expression |
---|---|
sqrtrval | ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2104 | . . . 4 ⊢ (𝑦 = 𝐴 → ((𝑥↑2) = 𝑦 ↔ (𝑥↑2) = 𝐴)) | |
2 | 1 | anbi1d 454 | . . 3 ⊢ (𝑦 = 𝐴 → (((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥) ↔ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) |
3 | 2 | riotabidv 5648 | . 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥)) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) |
4 | df-rsqrt 10562 | . 2 ⊢ √ = (𝑦 ∈ ℝ ↦ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝑦 ∧ 0 ≤ 𝑥))) | |
5 | reex 7573 | . . 3 ⊢ ℝ ∈ V | |
6 | riotaexg 5650 | . . 3 ⊢ (ℝ ∈ V → (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V) | |
7 | 5, 6 | ax-mp 7 | . 2 ⊢ (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥)) ∈ V |
8 | 3, 4, 7 | fvmpt 5416 | 1 ⊢ (𝐴 ∈ ℝ → (√‘𝐴) = (℩𝑥 ∈ ℝ ((𝑥↑2) = 𝐴 ∧ 0 ≤ 𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1296 ∈ wcel 1445 Vcvv 2633 class class class wbr 3867 ‘cfv 5049 ℩crio 5645 (class class class)co 5690 ℝcr 7446 0cc0 7447 ≤ cle 7620 2c2 8571 ↑cexp 10085 √csqrt 10560 |
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-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-cnex 7533 ax-resscn 7534 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rex 2376 df-v 2635 df-sbc 2855 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-rsqrt 10562 |
This theorem is referenced by: sqrt0 10568 resqrtcl 10593 rersqrtthlem 10594 sqrtsq 10608 |
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