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Mirrors > Home > ILE Home > Th. List > rpcxpsqrt | GIF version |
Description: The exponential function with exponent 1 / 2 exactly matches the square root function, and thus serves as a suitable generalization to other 𝑛-th roots and irrational roots. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 16-Jun-2024.) |
Ref | Expression |
---|---|
rpcxpsqrt | ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) = (√‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfre 9025 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
2 | rpcxpcl 13163 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ (1 / 2) ∈ ℝ) → (𝐴↑𝑐(1 / 2)) ∈ ℝ+) | |
3 | 1, 2 | mpan2 422 | . . 3 ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) ∈ ℝ+) |
4 | 3 | rpred 9581 | . 2 ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) ∈ ℝ) |
5 | rpre 9545 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
6 | rpge0 9551 | . . 3 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ 𝐴) | |
7 | 5, 6 | resqrtcld 11040 | . 2 ⊢ (𝐴 ∈ ℝ+ → (√‘𝐴) ∈ ℝ) |
8 | 3 | rpge0d 9585 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (𝐴↑𝑐(1 / 2))) |
9 | 5, 6 | sqrtge0d 11043 | . 2 ⊢ (𝐴 ∈ ℝ+ → 0 ≤ (√‘𝐴)) |
10 | ax-1cn 7804 | . . . . . 6 ⊢ 1 ∈ ℂ | |
11 | 2halves 9041 | . . . . . 6 ⊢ (1 ∈ ℂ → ((1 / 2) + (1 / 2)) = 1) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ ((1 / 2) + (1 / 2)) = 1 |
13 | 12 | oveq2i 5825 | . . . 4 ⊢ (𝐴↑𝑐((1 / 2) + (1 / 2))) = (𝐴↑𝑐1) |
14 | halfcn 9026 | . . . . 5 ⊢ (1 / 2) ∈ ℂ | |
15 | rpcxpadd 13165 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ (1 / 2) ∈ ℂ ∧ (1 / 2) ∈ ℂ) → (𝐴↑𝑐((1 / 2) + (1 / 2))) = ((𝐴↑𝑐(1 / 2)) · (𝐴↑𝑐(1 / 2)))) | |
16 | 14, 14, 15 | mp3an23 1308 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐((1 / 2) + (1 / 2))) = ((𝐴↑𝑐(1 / 2)) · (𝐴↑𝑐(1 / 2)))) |
17 | rpcxp1 13159 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐1) = 𝐴) | |
18 | 13, 16, 17 | 3eqtr3a 2211 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((𝐴↑𝑐(1 / 2)) · (𝐴↑𝑐(1 / 2))) = 𝐴) |
19 | 3 | rpcnd 9583 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) ∈ ℂ) |
20 | 19 | sqvald 10525 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((𝐴↑𝑐(1 / 2))↑2) = ((𝐴↑𝑐(1 / 2)) · (𝐴↑𝑐(1 / 2)))) |
21 | resqrtth 10908 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴)↑2) = 𝐴) | |
22 | 5, 6, 21 | syl2anc 409 | . . 3 ⊢ (𝐴 ∈ ℝ+ → ((√‘𝐴)↑2) = 𝐴) |
23 | 18, 20, 22 | 3eqtr4d 2197 | . 2 ⊢ (𝐴 ∈ ℝ+ → ((𝐴↑𝑐(1 / 2))↑2) = ((√‘𝐴)↑2)) |
24 | 4, 7, 8, 9, 23 | sq11d 10561 | 1 ⊢ (𝐴 ∈ ℝ+ → (𝐴↑𝑐(1 / 2)) = (√‘𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1332 ∈ wcel 2125 class class class wbr 3961 ‘cfv 5163 (class class class)co 5814 ℂcc 7709 ℝcr 7710 0cc0 7711 1c1 7712 + caddc 7714 · cmul 7716 ≤ cle 7892 / cdiv 8524 2c2 8863 ℝ+crp 9538 ↑cexp 10396 √csqrt 10873 ↑𝑐ccxp 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 ax-pre-suploc 7832 ax-addf 7833 ax-mulf 7834 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-disj 3939 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-of 6022 df-1st 6078 df-2nd 6079 df-recs 6242 df-irdg 6307 df-frec 6328 df-1o 6353 df-oadd 6357 df-er 6469 df-map 6584 df-pm 6585 df-en 6675 df-dom 6676 df-fin 6677 df-sup 6916 df-inf 6917 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-xneg 9657 df-xadd 9658 df-ioo 9774 df-ico 9776 df-icc 9777 df-fz 9891 df-fzo 10020 df-seqfrec 10323 df-exp 10397 df-fac 10577 df-bc 10599 df-ihash 10627 df-shft 10692 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-clim 11153 df-sumdc 11228 df-ef 11522 df-e 11523 df-rest 12292 df-topgen 12311 df-psmet 12326 df-xmet 12327 df-met 12328 df-bl 12329 df-mopn 12330 df-top 12335 df-topon 12348 df-bases 12380 df-ntr 12435 df-cn 12527 df-cnp 12528 df-tx 12592 df-cncf 12897 df-limced 12964 df-dvap 12965 df-relog 13118 df-rpcxp 13119 |
This theorem is referenced by: logsqrt 13182 sqrt2cxp2logb9e3 13231 |
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