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Theorem sqrlem2 14425

Description: Lemma for 01sqrex 14431. (Contributed by Mario Carneiro, 10-Jul-2013.)

**Hypotheses**
Ref	Expression
sqrlem1.1	⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
sqrlem1.2	⊢ 𝐵 = sup(𝑆, ℝ, < )

**Assertion**
Ref	Expression
sqrlem2	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆)

Distinct variable group: 𝑥,𝐴 Allowed substitution hints: 𝐵(𝑥) 𝑆(𝑥)

**Proof of Theorem sqrlem2**
Step	Hyp	Ref	Expression
1		simpl 483	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℝ⁺)
2		rpre 12236	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
3		rpgt0 12240	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
4		1re 10476	. . . . . 6 ⊢ 1 ∈ ℝ
5		lemul1 11329	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
6	4, 5	mp3an2 1439	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
7	2, 2, 3, 6	syl12anc 833	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
8	7	biimpa 477	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) ≤ (1 · 𝐴))
9		rpcn 12238	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
10	9	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℂ)
11		sqval 13319	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
12	11	eqcomd 2799	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 𝐴) = (𝐴↑2))
13	10, 12	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) = (𝐴↑2))
14	9	mulid2d 10494	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 · 𝐴) = 𝐴)
15	14	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (1 · 𝐴) = 𝐴)
16	8, 13, 15	3brtr3d 4987	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴↑2) ≤ 𝐴)
17		oveq1 7014	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2))
18	17	breq1d 4966	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) ≤ 𝐴 ↔ (𝐴↑2) ≤ 𝐴))
19		sqrlem1.1	. . 3 ⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
20	18, 19	elrab2 3616	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℝ⁺ ∧ (𝐴↑2) ≤ 𝐴))
21	1, 16, 20	sylanbrc 583	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 {crab 3107 class class class wbr 4956 (class class class)co 7007 supcsup 8740 ℂcc 10370 ℝcr 10371 0cc0 10372 1c1 10373 · cmul 10377 < clt 10510 ≤ cle 10511 2c2 11529 ℝ⁺crp 12228 ↑cexp 13267

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-er 8130 df-en 8348 df-dom 8349 df-sdom 8350 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-nn 11476 df-2 11537 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-seq 13208 df-exp 13268

This theorem is referenced by: sqrlem3 14426 sqrlem4 14427

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