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| Mirrors > Home > MPE Home > Th. List > 01sqrexlem5 | Structured version Visualization version GIF version | ||
| Description: Lemma for 01sqrex 15288. (Contributed by Mario Carneiro, 10-Jul-2013.) |
| Ref | Expression |
|---|---|
| 01sqrexlem1.1 | ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} |
| 01sqrexlem1.2 | ⊢ 𝐵 = sup(𝑆, ℝ, < ) |
| 01sqrexlem5.3 | ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} |
| Ref | Expression |
|---|---|
| 01sqrexlem5 | ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → ((𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣) ∧ (𝐵↑2) = sup(𝑇, ℝ, < ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 01sqrexlem1.1 | . . . . . . 7 ⊢ 𝑆 = {𝑥 ∈ ℝ+ ∣ (𝑥↑2) ≤ 𝐴} | |
| 2 | 1 | ssrab3 4082 | . . . . . 6 ⊢ 𝑆 ⊆ ℝ+ |
| 3 | 2 | sseli 3979 | . . . . 5 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ ℝ+) |
| 4 | 3 | rpge0d 13081 | . . . 4 ⊢ (𝑣 ∈ 𝑆 → 0 ≤ 𝑣) |
| 5 | 4 | rgen 3063 | . . 3 ⊢ ∀𝑣 ∈ 𝑆 0 ≤ 𝑣 |
| 6 | 01sqrexlem1.2 | . . . 4 ⊢ 𝐵 = sup(𝑆, ℝ, < ) | |
| 7 | 1, 6 | 01sqrexlem3 15283 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑆 ⊆ ℝ ∧ 𝑆 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑧 ∈ 𝑆 𝑧 ≤ 𝑣)) |
| 8 | 01sqrexlem5.3 | . . . 4 ⊢ 𝑇 = {𝑦 ∣ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑆 𝑦 = (𝑎 · 𝑏)} | |
| 9 | pm4.24 563 | . . . . 5 ⊢ (∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ↔ (∀𝑣 ∈ 𝑆 0 ≤ 𝑣 ∧ ∀𝑣 ∈ 𝑆 0 ≤ 𝑣)) | |
| 10 | 9 | 3anbi1i 1158 | . . . 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))) |
| 11 | 8, 10 | supmullem2 12239 | . . 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)) |
| 12 | 5, 7, 7, 11 | mp3an2i 1468 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝑇 ⊆ ℝ ∧ 𝑇 ≠ ∅ ∧ ∃𝑣 ∈ ℝ ∀𝑢 ∈ 𝑇 𝑢 ≤ 𝑣)) |
| 13 | 1, 6 | 01sqrexlem4 15284 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵 ∈ ℝ+ ∧ 𝐵 ≤ 1)) |
| 14 | rpre 13043 | . . . . . . 7 ⊢ (𝐵 ∈ ℝ+ → 𝐵 ∈ ℝ) | |
| 15 | 14 | adantr 480 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 ≤ 1) → 𝐵 ∈ ℝ) |
| 16 | 13, 15 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐵 ∈ ℝ) |
| 17 | 16 | recnd 11289 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → 𝐵 ∈ ℂ) |
| 18 | 17 | sqvald 14183 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵↑2) = (𝐵 · 𝐵)) |
| 19 | 6, 6 | oveq12i 7443 | . . . 4 ⊢ (𝐵 · 𝐵) = (sup(𝑆, ℝ, < ) · sup(𝑆, ℝ, < )) |
| 20 | 8, 10 | supmul 12240 | . . . . 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, < )) |
| 21 | 5, 7, 7, 20 | mp3an2i 1468 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (sup(𝑆, ℝ, < ) · sup(𝑆, ℝ, < )) = sup(𝑇, ℝ, < )) |
| 22 | 19, 21 | eqtrid 2789 | . . 3 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵 · 𝐵) = sup(𝑇, ℝ, < )) |
| 23 | 18, 22 | eqtrd 2777 | . 2 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝐴 ≤ 1) → (𝐵↑2) = sup(𝑇, ℝ, < )) |
| 24 | 12, 23 | jca 511 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3436 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 (class class class)co 7431 supcsup 9480 ℝcr 11154 0cc0 11155 1c1 11156 · cmul 11160 < clt 11295 ≤ cle 11296 2c2 12321 ℝ+crp 13034 ↑cexp 14102 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 |
| This theorem is referenced by: 01sqrexlem6 15286 |
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