01sqrexlem2 - Metamath Proof Explorer

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Theorem 01sqrexlem2 15205

Description: Lemma for 01sqrex 15211. (Contributed by Mario Carneiro, 10-Jul-2013.)

**Hypotheses**
Ref	Expression
01sqrexlem1.1	⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
01sqrexlem1.2	⊢ 𝐵 = sup(𝑆, ℝ, < )

**Assertion**
Ref	Expression
01sqrexlem2	⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆)

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

**Proof of Theorem 01sqrexlem2**
Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℝ⁺)
2		rpre 12951	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℝ)
3		rpgt0 12955	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 0 < 𝐴)
4		1re 11144	. . . . . 6 ⊢ 1 ∈ ℝ
5		lemul1 12007	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
6	4, 5	mp3an2 1452	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
7	2, 2, 3, 6	syl12anc 837	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴)))
8	7	biimpa 476	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) ≤ (1 · 𝐴))
9		rpcn 12953	. . . . 5 ⊢ (𝐴 ∈ ℝ⁺ → 𝐴 ∈ ℂ)
10	9	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℂ)
11		sqval 14076	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
12	11	eqcomd 2742	. . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 𝐴) = (𝐴↑2))
13	10, 12	syl 17	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) = (𝐴↑2))
14	9	mullidd 11163	. . . 4 ⊢ (𝐴 ∈ ℝ⁺ → (1 · 𝐴) = 𝐴)
15	14	adantr 480	. . 3 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (1 · 𝐴) = 𝐴)
16	8, 13, 15	3brtr3d 5116	. 2 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → (𝐴↑2) ≤ 𝐴)
17		oveq1 7374	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2))
18	17	breq1d 5095	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) ≤ 𝐴 ↔ (𝐴↑2) ≤ 𝐴))
19		01sqrexlem1.1	. . 3 ⊢ 𝑆 = {𝑥 ∈ ℝ⁺ ∣ (𝑥↑2) ≤ 𝐴}
20	18, 19	elrab2 3637	. 2 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℝ⁺ ∧ (𝐴↑2) ≤ 𝐴))
21	1, 16, 20	sylanbrc 584	1 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3389 class class class wbr 5085 (class class class)co 7367 supcsup 9353 ℂcc 11036 ℝcr 11037 0cc0 11038 1c1 11039 · cmul 11043 < clt 11179 ≤ cle 11180 2c2 12236 ℝ⁺crp 12942 ↑cexp 14023

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-rp 12943 df-seq 13964 df-exp 14024

This theorem is referenced by: 01sqrexlem3 15206 01sqrexlem4 15207

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