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Mirrors > Home > ILE Home > Th. List > resqrexlem1arp | GIF version |
Description: Lemma for resqrex 10766. 1 + 𝐴 is a positive real (expressed in a way that will help apply seqf 10202 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
Ref | Expression |
---|---|
resqrexlem1arp.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resqrexlem1arp.agt0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
resqrexlem1arp | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 7749 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 1 ∈ ℝ) | |
2 | resqrexlem1arp.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 2 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ) |
4 | 1, 3 | readdcld 7763 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 + 𝐴) ∈ ℝ) |
5 | 0lt1 7857 | . . . . . 6 ⊢ 0 < 1 | |
6 | 5 | a1i 9 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 1) |
7 | resqrexlem1arp.agt0 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐴) | |
8 | 7 | adantr 274 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝐴) |
9 | addgtge0 8180 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 < 1 ∧ 0 ≤ 𝐴)) → 0 < (1 + 𝐴)) | |
10 | 1, 3, 6, 8, 9 | syl22anc 1202 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < (1 + 𝐴)) |
11 | 4, 10 | elrpd 9449 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 + 𝐴) ∈ ℝ+) |
12 | fvconst2g 5602 | . . 3 ⊢ (((1 + 𝐴) ∈ ℝ+ ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) = (1 + 𝐴)) | |
13 | 11, 12 | sylancom 416 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) = (1 + 𝐴)) |
14 | 13, 11 | eqeltrd 2194 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 {csn 3497 class class class wbr 3899 × cxp 4507 ‘cfv 5093 (class class class)co 5742 ℝcr 7587 0cc0 7588 1c1 7589 + caddc 7591 < clt 7768 ≤ cle 7769 ℕcn 8688 ℝ+crp 9409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-i2m1 7693 ax-0lt1 7694 ax-0id 7696 ax-rnegex 7697 ax-pre-ltwlin 7701 ax-pre-ltadd 7704 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-rp 9410 |
This theorem is referenced by: resqrexlemf 10747 resqrexlemf1 10748 resqrexlemfp1 10749 |
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