| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > pntlemd | 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, 𝐴 is C^*, 𝐵 is c1, 𝐿 is λ, 𝐷 is c2, and 𝐹 is c3. (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))) |
| Ref | Expression |
|---|---|
| pntlemd | ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ioossre 13448 | . . . 4 ⊢ (0(,)1) ⊆ ℝ | |
| 2 | pntlem1.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
| 3 | 1, 2 | sselid 3981 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
| 4 | eliooord 13446 | . . . . 5 ⊢ (𝐿 ∈ (0(,)1) → (0 < 𝐿 ∧ 𝐿 < 1)) | |
| 5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (0 < 𝐿 ∧ 𝐿 < 1)) |
| 6 | 5 | simpld 494 | . . 3 ⊢ (𝜑 → 0 < 𝐿) |
| 7 | 3, 6 | elrpd 13074 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
| 8 | pntlem1.d | . . 3 ⊢ 𝐷 = (𝐴 + 1) | |
| 9 | pntlem1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
| 10 | 1rp 13038 | . . . 4 ⊢ 1 ∈ ℝ+ | |
| 11 | rpaddcl 13057 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
| 12 | 9, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
| 13 | 8, 12 | eqeltrid 2845 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
| 14 | pntlem1.f | . . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
| 15 | 1re 11261 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
| 16 | ltaddrp 13072 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
| 17 | 15, 9, 16 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
| 18 | 9 | rpcnd 13079 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 19 | ax-1cn 11213 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
| 20 | addcom 11447 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
| 21 | 18, 19, 20 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
| 22 | 8, 21 | eqtrid 2789 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
| 23 | 17, 22 | breqtrrd 5171 | . . . . . 6 ⊢ (𝜑 → 1 < 𝐷) |
| 24 | 13 | recgt1d 13091 | . . . . . 6 ⊢ (𝜑 → (1 < 𝐷 ↔ (1 / 𝐷) < 1)) |
| 25 | 23, 24 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (1 / 𝐷) < 1) |
| 26 | 13 | rprecred 13088 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
| 27 | difrp 13073 | . . . . . 6 ⊢ (((1 / 𝐷) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) | |
| 28 | 26, 15, 27 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) |
| 29 | 25, 28 | mpbid 232 | . . . 4 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℝ+) |
| 30 | 3nn0 12544 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
| 31 | 2nn 12339 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
| 32 | 30, 31 | decnncl 12753 | . . . . . . . 8 ⊢ ;32 ∈ ℕ |
| 33 | nnrp 13046 | . . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
| 34 | 32, 33 | ax-mp 5 | . . . . . . 7 ⊢ ;32 ∈ ℝ+ |
| 35 | pntlem1.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
| 36 | rpmulcl 13058 | . . . . . . 7 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
| 37 | 34, 35, 36 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
| 38 | 7, 37 | rpdivcld 13094 | . . . . 5 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ+) |
| 39 | 2z 12649 | . . . . . 6 ⊢ 2 ∈ ℤ | |
| 40 | rpexpcl 14121 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+) | |
| 41 | 13, 39, 40 | sylancl 586 | . . . . 5 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ+) |
| 42 | 38, 41 | rpdivcld 13094 | . . . 4 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+) |
| 43 | 29, 42 | rpmulcld 13093 | . . 3 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ∈ ℝ+) |
| 44 | 14, 43 | eqeltrid 2845 | . 2 ⊢ (𝜑 → 𝐹 ∈ ℝ+) |
| 45 | 7, 13, 44 | 3jca 1129 | 1 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 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 < clt 11295 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 3c3 12322 ℤcz 12613 ;cdc 12733 ℝ+crp 13034 (,)cioo 13387 ↑cexp 14102 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-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 |
| 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-1st 8014 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-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-seq 14043 df-exp 14103 |
| This theorem is referenced by: pntlemc 27639 pntlema 27640 pntlemb 27641 pntlemq 27645 pntlemr 27646 pntlemj 27647 pntlemf 27649 pntlemo 27651 pntleml 27655 |
| Copyright terms: Public domain | W3C validator |