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Mirrors > Home > MPE Home > Th. List > sqrlem1 | Structured version Visualization version GIF version |
Description: Lemma for 01sqrex 14603. (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 |
---|---|
sqrlem1 | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7157 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2)) | |
2 | 1 | breq1d 5068 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥↑2) ≤ 𝐴 ↔ (𝑦↑2) ≤ 𝐴)) |
3 | sqrlem1.1 | . . . 4 ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} | |
4 | 2, 3 | elrab2 3682 | . . 3 ⊢ (𝑦 ∈ 𝑆 ↔ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) |
5 | simprr 771 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → (𝑦↑2) ≤ 𝐴) | |
6 | simplr 767 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → 𝐴 ≤ 1) | |
7 | rpre 12391 | . . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ+ → 𝑦 ∈ ℝ) | |
8 | 7 | ad2antrl 726 | . . . . . . . . 9 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → 𝑦 ∈ ℝ) |
9 | 8 | resqcld 13605 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → (𝑦↑2) ∈ ℝ) |
10 | rpre 12391 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) | |
11 | 10 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → 𝐴 ∈ ℝ) |
12 | 1re 10635 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
13 | letr 10728 | . . . . . . . . 9 ⊢ (((𝑦↑2) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (((𝑦↑2) ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝑦↑2) ≤ 1)) | |
14 | 12, 13 | mp3an3 1446 | . . . . . . . 8 ⊢ (((𝑦↑2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((𝑦↑2) ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝑦↑2) ≤ 1)) |
15 | 9, 11, 14 | syl2anc 586 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → (((𝑦↑2) ≤ 𝐴 ∧ 𝐴 ≤ 1) → (𝑦↑2) ≤ 1)) |
16 | 5, 6, 15 | mp2and 697 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → (𝑦↑2) ≤ 1) |
17 | sq1 13552 | . . . . . 6 ⊢ (1↑2) = 1 | |
18 | 16, 17 | breqtrrdi 5100 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → (𝑦↑2) ≤ (1↑2)) |
19 | rpge0 12396 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ+ → 0 ≤ 𝑦) | |
20 | 19 | ad2antrl 726 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → 0 ≤ 𝑦) |
21 | 0le1 11157 | . . . . . . 7 ⊢ 0 ≤ 1 | |
22 | le2sq 13493 | . . . . . . 7 ⊢ (((𝑦 ∈ ℝ ∧ 0 ≤ 𝑦) ∧ (1 ∈ ℝ ∧ 0 ≤ 1)) → (𝑦 ≤ 1 ↔ (𝑦↑2) ≤ (1↑2))) | |
23 | 12, 21, 22 | mpanr12 703 | . . . . . 6 ⊢ ((𝑦 ∈ ℝ ∧ 0 ≤ 𝑦) → (𝑦 ≤ 1 ↔ (𝑦↑2) ≤ (1↑2))) |
24 | 8, 20, 23 | syl2anc 586 | . . . . 5 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → (𝑦 ≤ 1 ↔ (𝑦↑2) ≤ (1↑2))) |
25 | 18, 24 | mpbird 259 | . . . 4 ⊢ (((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) ∧ (𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴)) → 𝑦 ≤ 1) |
26 | 25 | ex 415 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ((𝑦 ∈ ℝ+ ∧ (𝑦↑2) ≤ 𝐴) → 𝑦 ≤ 1)) |
27 | 4, 26 | syl5bi 244 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑦 ∈ 𝑆 → 𝑦 ≤ 1)) |
28 | 27 | ralrimiv 3181 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 class class class wbr 5058 (class class class)co 7150 supcsup 8898 ℝcr 10530 0cc0 10531 1c1 10532 < clt 10669 ≤ cle 10670 2c2 11686 ℝ+crp 12383 ↑cexp 13423 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 |
This theorem is referenced by: sqrlem3 14598 sqrlem4 14599 |
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