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Theorem sqrlem2 15024

Description: Lemma for 01sqrex 15030. (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 12808	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
3		rpgt0 12812	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
4		1re 11045	. . . . . 6 ⊢ 1 ∈ ℝ
5		lemul1 11897	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
6	4, 5	mp3an2 1448	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
7	2, 2, 3, 6	syl12anc 834	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
8	7	biimpa 477	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) ≤ (1 · 𝐴))
9		rpcn 12810	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
10	9	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℂ)
11		sqval 13905	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
12	11	eqcomd 2743	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 𝐴) = (𝐴↑2))
13	10, 12	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) = (𝐴↑2))
14	9	mulid2d 11063	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 · 𝐴) = 𝐴)
15	14	adantr 481	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (1 · 𝐴) = 𝐴)
16	8, 13, 15	3brtr3d 5116	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴↑2) ≤ 𝐴)
17		oveq1 7320	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2))
18	17	breq1d 5095	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) ≤ 𝐴 ↔ (𝐴↑2) ≤ 𝐴))
19		sqrlem1.1	. . 3 ⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
20	18, 19	elrab2 3636	. 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 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 {crab 3404 class class class wbr 5085 (class class class)co 7313 supcsup 9267 ℂcc 10939 ℝcr 10940 0cc0 10941 1c1 10942 · cmul 10946 < clt 11079 ≤ cle 11080 2c2 12098 ℝ⁺crp 12800 ↑cexp 13852

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4849 df-iun 4937 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-om 7756 df-2nd 7875 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-er 8544 df-en 8780 df-dom 8781 df-sdom 8782 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-nn 12044 df-2 12106 df-n0 12304 df-z 12390 df-uz 12653 df-rp 12801 df-seq 13792 df-exp 13853

This theorem is referenced by: sqrlem3 15025 sqrlem4 15026

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