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| Mirrors > Home > MPE Home > Th. List > pntlemc | Structured version Visualization version GIF version | ||
| Description: Lemma for pnt 27658. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑈 is α, 𝐸 is ε, and 𝐾 is K. (Contributed by Mario Carneiro, 13-Apr-2016.) |
| Ref | Expression |
|---|---|
| pntlem1.r | ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) |
| pntlem1.a | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
| pntlem1.b | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
| pntlem1.l | ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |
| pntlem1.d | ⊢ 𝐷 = (𝐴 + 1) |
| pntlem1.f | ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) |
| pntlem1.u | ⊢ (𝜑 → 𝑈 ∈ ℝ+) |
| pntlem1.u2 | ⊢ (𝜑 → 𝑈 ≤ 𝐴) |
| pntlem1.e | ⊢ 𝐸 = (𝑈 / 𝐷) |
| pntlem1.k | ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) |
| Ref | Expression |
|---|---|
| pntlemc | ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pntlem1.e | . . 3 ⊢ 𝐸 = (𝑈 / 𝐷) | |
| 2 | pntlem1.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ+) | |
| 3 | pntlem1.r | . . . . . 6 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) | |
| 4 | pntlem1.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
| 5 | pntlem1.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
| 6 | pntlem1.l | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
| 7 | pntlem1.d | . . . . . 6 ⊢ 𝐷 = (𝐴 + 1) | |
| 8 | pntlem1.f | . . . . . 6 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
| 9 | 3, 4, 5, 6, 7, 8 | pntlemd 27638 | . . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
| 10 | 9 | simp2d 1144 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
| 11 | 2, 10 | rpdivcld 13094 | . . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ+) |
| 12 | 1, 11 | eqeltrid 2845 | . 2 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
| 13 | pntlem1.k | . . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) | |
| 14 | 5, 12 | rpdivcld 13094 | . . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ+) |
| 15 | 14 | rpred 13077 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ) |
| 16 | 15 | rpefcld 16141 | . . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ+) |
| 17 | 13, 16 | eqeltrid 2845 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
| 18 | 12 | rpred 13077 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
| 19 | 12 | rpgt0d 13080 | . . . 4 ⊢ (𝜑 → 0 < 𝐸) |
| 20 | 2 | rpred 13077 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
| 21 | 4 | rpred 13077 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 22 | 10 | rpred 13077 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
| 23 | pntlem1.u2 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴) | |
| 24 | 21 | ltp1d 12198 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
| 25 | 24, 7 | breqtrrdi 5185 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷) |
| 26 | 20, 21, 22, 23, 25 | lelttrd 11419 | . . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷) |
| 27 | 10 | rpcnd 13079 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
| 28 | 27 | mulridd 11278 | . . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷) |
| 29 | 26, 28 | breqtrrd 5171 | . . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1)) |
| 30 | 1red 11262 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
| 31 | 20, 30, 10 | ltdivmuld 13128 | . . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1))) |
| 32 | 29, 31 | mpbird 257 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1) |
| 33 | 1, 32 | eqbrtrid 5178 | . . . 4 ⊢ (𝜑 → 𝐸 < 1) |
| 34 | 0xr 11308 | . . . . 5 ⊢ 0 ∈ ℝ* | |
| 35 | 1xr 11320 | . . . . 5 ⊢ 1 ∈ ℝ* | |
| 36 | elioo2 13428 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))) | |
| 37 | 34, 35, 36 | mp2an 692 | . . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)) |
| 38 | 18, 19, 33, 37 | syl3anbrc 1344 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) |
| 39 | efgt1 16152 | . . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ+ → 1 < (exp‘(𝐵 / 𝐸))) | |
| 40 | 14, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸))) |
| 41 | 40, 13 | breqtrrdi 5185 | . . 3 ⊢ (𝜑 → 1 < 𝐾) |
| 42 | 1re 11261 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 43 | ltaddrp 13072 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
| 44 | 42, 4, 43 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
| 45 | 2 | rpcnne0d 13086 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0)) |
| 46 | divid 11953 | . . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1) | |
| 47 | 45, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1) |
| 48 | 4 | rpcnd 13079 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 49 | ax-1cn 11213 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
| 50 | addcom 11447 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
| 51 | 48, 49, 50 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
| 52 | 7, 51 | eqtrid 2789 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
| 53 | 44, 47, 52 | 3brtr4d 5175 | . . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷) |
| 54 | 20, 2, 10, 53 | ltdiv23d 13144 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈) |
| 55 | 1, 54 | eqbrtrid 5178 | . . . 4 ⊢ (𝜑 → 𝐸 < 𝑈) |
| 56 | difrp 13073 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) | |
| 57 | 18, 20, 56 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) |
| 58 | 55, 57 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
| 59 | 38, 41, 58 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
| 60 | 12, 17, 59 | 3jca 1129 | 1 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 2c2 12321 3c3 12322 ;cdc 12733 ℝ+crp 13034 (,)cioo 13387 ↑cexp 14102 expce 16097 ψcchp 27136 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 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-int 4947 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-se 5638 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-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 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-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-ioo 13391 df-ico 13393 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 |
| This theorem is referenced by: pntlema 27640 pntlemb 27641 pntlemg 27642 pntlemh 27643 pntlemq 27645 pntlemr 27646 pntlemj 27647 pntlemi 27648 pntlemf 27649 pntlemo 27651 pntleme 27652 pntlemp 27654 |
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