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Mirrors > Home > ILE Home > Th. List > resqrexlemf | GIF version |
Description: Lemma for resqrex 10979. The sequence is a function. (Contributed by Mario Carneiro and Jim Kingdon, 27-Jul-2021.) (Revised by Jim Kingdon, 16-Oct-2022.) |
Ref | Expression |
---|---|
resqrexlemex.seq | ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) |
resqrexlemex.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resqrexlemex.agt0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
resqrexlemf | ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 9511 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 9228 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
3 | resqrexlemex.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
4 | resqrexlemex.agt0 | . . . 4 ⊢ (𝜑 → 0 ≤ 𝐴) | |
5 | 3, 4 | resqrexlem1arp 10958 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℕ) → ((ℕ × {(1 + 𝐴)})‘𝑎) ∈ ℝ+) |
6 | 3, 4 | resqrexlemp1rp 10959 | . . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+)) → (𝑎(𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2))𝑏) ∈ ℝ+) |
7 | 1, 2, 5, 6 | seqf 10406 | . 2 ⊢ (𝜑 → seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})):ℕ⟶ℝ+) |
8 | resqrexlemex.seq | . . 3 ⊢ 𝐹 = seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})) | |
9 | 8 | feq1i 5338 | . 2 ⊢ (𝐹:ℕ⟶ℝ+ ↔ seq1((𝑦 ∈ ℝ+, 𝑧 ∈ ℝ+ ↦ ((𝑦 + (𝐴 / 𝑦)) / 2)), (ℕ × {(1 + 𝐴)})):ℕ⟶ℝ+) |
10 | 7, 9 | sylibr 133 | 1 ⊢ (𝜑 → 𝐹:ℕ⟶ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 {csn 3581 class class class wbr 3987 × cxp 4607 ⟶wf 5192 (class class class)co 5851 ∈ cmpo 5853 ℝcr 7762 0cc0 7763 1c1 7764 + caddc 7766 ≤ cle 7944 / cdiv 8578 ℕcn 8867 2c2 8918 ℝ+crp 9599 seqcseq 10390 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7854 ax-resscn 7855 ax-1cn 7856 ax-1re 7857 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-mulrcl 7862 ax-addcom 7863 ax-mulcom 7864 ax-addass 7865 ax-mulass 7866 ax-distr 7867 ax-i2m1 7868 ax-0lt1 7869 ax-1rid 7870 ax-0id 7871 ax-rnegex 7872 ax-precex 7873 ax-cnre 7874 ax-pre-ltirr 7875 ax-pre-ltwlin 7876 ax-pre-lttrn 7877 ax-pre-apti 7878 ax-pre-ltadd 7879 ax-pre-mulgt0 7880 ax-pre-mulext 7881 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7945 df-mnf 7946 df-xr 7947 df-ltxr 7948 df-le 7949 df-sub 8081 df-neg 8082 df-reap 8483 df-ap 8490 df-div 8579 df-inn 8868 df-2 8926 df-n0 9125 df-z 9202 df-uz 9477 df-rp 9600 df-seqfrec 10391 |
This theorem is referenced by: resqrexlemfp1 10962 resqrexlemover 10963 resqrexlemdec 10964 resqrexlemdecn 10965 resqrexlemlo 10966 resqrexlemcalc1 10967 resqrexlemcalc2 10968 resqrexlemcalc3 10969 resqrexlemnmsq 10970 resqrexlemnm 10971 resqrexlemcvg 10972 resqrexlemgt0 10973 resqrexlemoverl 10974 resqrexlemglsq 10975 resqrexlemga 10976 resqrexlemsqa 10977 |
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