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Mirrors > Home > MPE Home > Th. List > sqrlem2 | Structured version Visualization version GIF version |
Description: Lemma for 01sqrex 14601. (Contributed by Mario Carneiro, 10-Jul-2013.) |
Ref | Expression |
---|---|
sqrlem1.1 | ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} |
sqrlem1.2 | ⊢ 𝐵 = sup(𝑆, ℝ, < ) |
Ref | Expression |
---|---|
sqrlem2 | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℝ+) | |
2 | rpre 12389 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
3 | rpgt0 12393 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) | |
4 | 1re 10633 | . . . . . 6 ⊢ 1 ∈ ℝ | |
5 | lemul1 11484 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 1 ∈ ℝ ∧ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) → (𝐴 ≤ 1 ↔ (𝐴 · 𝐴) ≤ (1 · 𝐴))) | |
6 | 4, 5 | mp3an2 1443 | . . . . 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 479 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) ≤ (1 · 𝐴)) |
9 | rpcn 12391 | . . . . 5 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ) | |
10 | 9 | adantr 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ ℂ) |
11 | sqval 13473 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴)) | |
12 | 11 | eqcomd 2825 | . . . 4 ⊢ (𝐴 ∈ ℂ → (𝐴 · 𝐴) = (𝐴↑2)) |
13 | 10, 12 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐴 · 𝐴) = (𝐴↑2)) |
14 | 9 | mulid2d 10651 | . . . 4 ⊢ (𝐴 ∈ ℝ+ → (1 · 𝐴) = 𝐴) |
15 | 14 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (1 · 𝐴) = 𝐴) |
16 | 8, 13, 15 | 3brtr3d 5088 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐴↑2) ≤ 𝐴) |
17 | oveq1 7155 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥↑2) = (𝐴↑2)) | |
18 | 17 | breq1d 5067 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥↑2) ≤ 𝐴 ↔ (𝐴↑2) ≤ 𝐴)) |
19 | sqrlem1.1 | . . 3 ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} | |
20 | 18, 19 | elrab2 3681 | . 2 ⊢ (𝐴 ∈ 𝑆 ↔ (𝐴 ∈ ℝ+ ∧ (𝐴↑2) ≤ 𝐴)) |
21 | 1, 16, 20 | sylanbrc 585 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 {crab 3140 class class class wbr 5057 (class class class)co 7148 supcsup 8896 ℂcc 10527 ℝcr 10528 0cc0 10529 1c1 10530 · cmul 10534 < clt 10667 ≤ cle 10668 2c2 11684 ℝ+crp 12381 ↑cexp 13421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-seq 13362 df-exp 13422 |
This theorem is referenced by: sqrlem3 14596 sqrlem4 14597 |
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