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Mirrors > Home > ILE Home > Th. List > resqrexlem1arp | GIF version |
Description: Lemma for resqrex 9912. 1 + 𝐴 is a positive real (expressed in a way that will help apply iseqf 9444 and similar theorems). (Contributed by Jim Kingdon, 28-Jul-2021.) |
Ref | Expression |
---|---|
resqrexlemex.seq | ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)}), ℝ+) |
resqrexlemex.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resqrexlemex.agt0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
resqrexlem1arp | ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1red 7134 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 1 ∈ ℝ) | |
2 | resqrexlemex.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | 2 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ) |
4 | 1, 3 | readdcld 7148 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 + 𝐴) ∈ ℝ) |
5 | 0lt1 7236 | . . . . . 6 ⊢ 0 < 1 | |
6 | 5 | a1i 9 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < 1) |
7 | resqrexlemex.agt0 | . . . . . 6 ⊢ (𝜑 → 0 ≤ 𝐴) | |
8 | 7 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝐴) |
9 | addgtge0 7554 | . . . . 5 ⊢ (((1 ∈ ℝ ∧ 𝐴 ∈ ℝ) ∧ (0 < 1 ∧ 0 ≤ 𝐴)) → 0 < (1 + 𝐴)) | |
10 | 1, 3, 6, 8, 9 | syl22anc 1170 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 0 < (1 + 𝐴)) |
11 | 4, 10 | elrpd 8771 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (1 + 𝐴) ∈ ℝ+) |
12 | fvconst2g 5396 | . . 3 ⊢ (((1 + 𝐴) ∈ ℝ+ ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) = (1 + 𝐴)) | |
13 | 11, 12 | sylancom 411 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) = (1 + 𝐴)) |
14 | 13, 11 | eqeltrd 2155 | 1 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑁) ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1284 ∈ wcel 1433 {csn 3398 class class class wbr 3785 × cxp 4361 ‘cfv 4922 (class class class)co 5532 ↦ cmpt2 5534 ℝcr 6980 0cc0 6981 1c1 6982 + caddc 6984 < clt 7153 ≤ cle 7154 / cdiv 7760 ℕcn 8039 2c2 8089 ℝ+crp 8734 seqcseq 9431 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-i2m1 7081 ax-0lt1 7082 ax-0id 7084 ax-rnegex 7085 ax-pre-ltwlin 7089 ax-pre-ltadd 7092 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-ov 5535 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-rp 8735 |
This theorem is referenced by: resqrexlemf 9893 resqrexlemf1 9894 resqrexlemfp1 9895 |
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