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Mirrors > Home > MPE Home > Th. List > sqrlem5 | Structured version Visualization version GIF version |
Description: Lemma for 01sqrex 14328. (Contributed by Mario Carneiro, 10-Jul-2013.) |
Ref | Expression |
---|---|
sqrlem1.1 | ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} |
sqrlem1.2 | ⊢ 𝐵 = sup(𝑆, ℝ, < ) |
sqrlem5.3 | ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} |
Ref | Expression |
---|---|
sqrlem5 | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣) ∧ (𝐵↑2) = sup(𝑇, ℝ, < ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrlem1.1 | . . . . . . . 8 ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} | |
2 | ssrab2 3882 | . . . . . . . 8 ⊢ {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} ⊆ ℝ+ | |
3 | 1, 2 | eqsstri 3830 | . . . . . . 7 ⊢ 𝑆 ⊆ ℝ+ |
4 | 3 | sseli 3793 | . . . . . 6 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ ℝ+) |
5 | 4 | rpge0d 12118 | . . . . 5 ⊢ (𝑣 ∈ 𝑆 → 0 ≤ 𝑣) |
6 | 5 | rgen 3102 | . . . 4 ⊢ ∀𝑣 ∈ 𝑆 0 ≤ 𝑣 |
7 | 6 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ∀𝑣 ∈ 𝑆 0 ≤ 𝑣) |
8 | sqrlem1.2 | . . . 4 ⊢ 𝐵 = sup(𝑆, ℝ, < ) | |
9 | 1, 8 | sqrlem3 14323 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣)) |
10 | sqrlem5.3 | . . . 4 ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} | |
11 | pm4.24 560 | . . . . 5 ⊢ (∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ↔ (∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ∧ ∀𝑣 ∈ 𝑆 0 ≤ 𝑣)) | |
12 | 11 | 3anbi1i 1197 | . . . 4 ⊢ ((∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣) ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣)) ↔ ((∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ∧ ∀𝑣 ∈ 𝑆 0 ≤ 𝑣) ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣) ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣))) |
13 | 10, 12 | supmullem2 11285 | . . 3 ⊢ ((∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣) ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣)) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣)) |
14 | 7, 9, 9, 13 | syl3anc 1491 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣)) |
15 | 1, 8 | sqrlem4 14324 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵 ∈ ℝ+ ∧ 𝐵 ≤ 1)) |
16 | rpre 12079 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
17 | 16 | adantr 473 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≤ 1) → 𝐵 ∈ ℝ) |
18 | 15, 17 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐵 ∈ ℝ) |
19 | 18 | recnd 10356 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐵 ∈ ℂ) |
20 | 19 | sqvald 13256 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵↑2) = (𝐵 · 𝐵)) |
21 | 8, 8 | oveq12i 6889 | . . . 4 ⊢ (𝐵 · 𝐵) = (sup(𝑆, ℝ, < ) · sup(𝑆, ℝ, < )) |
22 | 10, 12 | supmul 11286 | . . . . 5 ⊢ ((∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣) ∧ (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣)) → (sup(𝑆, ℝ, < ) · sup(𝑆, ℝ, < )) = sup(𝑇, ℝ, < )) |
23 | 7, 9, 9, 22 | syl3anc 1491 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (sup(𝑆, ℝ, < ) · sup(𝑆, ℝ, < )) = sup(𝑇, ℝ, < )) |
24 | 21, 23 | syl5eq 2844 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵 · 𝐵) = sup(𝑇, ℝ, < )) |
25 | 20, 24 | eqtrd 2832 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵↑2) = sup(𝑇, ℝ, < )) |
26 | 14, 25 | jca 508 | 1 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣) ∧ (𝐵↑2) = sup(𝑇, ℝ, < ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {cab 2784 ≠ wne 2970 ∀wral 3088 ∃wrex 3089 {crab 3092 ⊆ wss 3768 ∅c0 4114 class class class wbr 4842 (class class class)co 6877 supcsup 8587 ℝcr 10222 0cc0 10223 1c1 10224 · cmul 10228 < clt 10362 ≤ cle 10363 2c2 11365 ℝ+crp 12071 ↑cexp 13111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-cnex 10279 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 ax-pre-mulgt0 10300 ax-pre-sup 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-pss 3784 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-tp 4372 df-op 4374 df-uni 4628 df-iun 4711 df-br 4843 df-opab 4905 df-mpt 4922 df-tr 4945 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5897 df-ord 5943 df-on 5944 df-lim 5945 df-suc 5946 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-riota 6838 df-ov 6880 df-oprab 6881 df-mpt2 6882 df-om 7299 df-2nd 7401 df-wrecs 7644 df-recs 7706 df-rdg 7744 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-sup 8589 df-pnf 10364 df-mnf 10365 df-xr 10366 df-ltxr 10367 df-le 10368 df-sub 10557 df-neg 10558 df-div 10976 df-nn 11312 df-2 11373 df-n0 11578 df-z 11664 df-uz 11928 df-rp 12072 df-seq 13053 df-exp 13112 |
This theorem is referenced by: sqrlem6 14326 |
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