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Mirrors > Home > MPE Home > Th. List > sqrlem3 | Structured version Visualization version GIF version |
Description: Lemma for 01sqrex 13990. (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 |
---|---|
sqrlem3 | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrlem1.1 | . . . 4 ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} | |
2 | ssrab2 3687 | . . . . 5 ⊢ {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} ⊆ ℝ+ | |
3 | rpssre 11843 | . . . . 5 ⊢ ℝ+ ⊆ ℝ | |
4 | 2, 3 | sstri 3612 | . . . 4 ⊢ {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} ⊆ ℝ |
5 | 1, 4 | eqsstri 3635 | . . 3 ⊢ 𝑆 ⊆ ℝ |
6 | 5 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝑆 ⊆ ℝ) |
7 | sqrlem1.2 | . . . 4 ⊢ 𝐵 = sup(𝑆, ℝ, < ) | |
8 | 1, 7 | sqrlem2 13984 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐴 ∈ 𝑆) |
9 | ne0i 3921 | . . 3 ⊢ (𝐴 ∈ 𝑆 → 𝑆 ≠ ∅) | |
10 | 8, 9 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝑆 ≠ ∅) |
11 | 1re 10039 | . . 3 ⊢ 1 ∈ ℝ | |
12 | 1, 7 | sqrlem1 13983 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∀𝑦 ∈ 𝑆 𝑦 ≤ 1) |
13 | breq2 4657 | . . . . 5 ⊢ (𝑧 = 1 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 1)) | |
14 | 13 | ralbidv 2986 | . . . 4 ⊢ (𝑧 = 1 → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ 1)) |
15 | 14 | rspcev 3309 | . . 3 ⊢ ((1 ∈ ℝ ∧ ∀𝑦 ∈ 𝑆 𝑦 ≤ 1) → ∃𝑧 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧) |
16 | 11, 12, 15 | sylancr 695 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∃𝑧 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧) |
17 | 6, 10, 16 | 3jca 1242 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 (class class class)co 6650 supcsup 8346 ℝcr 9935 1c1 9937 < clt 10074 ≤ cle 10075 2c2 11070 ℝ+crp 11832 ↑cexp 12860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 |
This theorem is referenced by: sqrlem4 13986 sqrlem5 13987 sqrlem6 13988 sqrlem7 13989 |
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