divsqrtid - Mathbox for Thierry Arnoux

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Theorem divsqrtid 34061

Description: A real number divided by its square root. (Contributed by Thierry Arnoux, 1-Jan-2022.)

**Assertion**
Ref	Expression
divsqrtid	⊢ (𝐴 ∈ ℝ⁺ → (𝐴 / (√‘𝐴)) = (√‘𝐴))

**Proof of Theorem divsqrtid**
Step	Hyp	Ref	Expression
1		rpre 12978	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
2		rpge0 12983	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 0 ≤ 𝐴)
3		remsqsqrt 15199	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → ((√‘𝐴) · (√‘𝐴)) = 𝐴)
4	1, 2, 3	syl2anc 583	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → ((√‘𝐴) · (√‘𝐴)) = 𝐴)
5	4	oveq1d 7416	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((√‘𝐴) · (√‘𝐴)) / (√‘𝐴)) = (𝐴 / (√‘𝐴)))
6	1	recnd 11238	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
7	6	sqrtcld 15380	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (√‘𝐴) ∈ ℂ)
8		rpsqrtcl 15207	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (√‘𝐴) ∈ ℝ⁺)
9	8	rpne0d 13017	. . 3 ⊢ (𝐴 ∈ ℝ⁺ → (√‘𝐴) ≠ 0)
10	7, 7, 9	divcan4d 11992	. 2 ⊢ (𝐴 ∈ ℝ⁺ → (((√‘𝐴) · (√‘𝐴)) / (√‘𝐴)) = (√‘𝐴))
11	5, 10	eqtr3d 2766	1 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 / (√‘𝐴)) = (√‘𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 ‘cfv 6533 (class class class)co 7401 ℝcr 11104 0cc0 11105 · cmul 11110 ≤ cle 11245 / cdiv 11867 ℝ⁺crp 12970 √csqrt 15176

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179

This theorem is referenced by: hgt750leme 34125

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