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Mirrors > Home > HSE Home > Th. List > opsqrlem4 | Structured version Visualization version GIF version |
Description: Lemma for opsqri . (Contributed by NM, 17-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
opsqrlem2.1 | ⊢ 𝑇 ∈ HrmOp |
opsqrlem2.2 | ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) |
opsqrlem2.3 | ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) |
Ref | Expression |
---|---|
opsqrlem4 | ⊢ 𝐹:ℕ⟶HrmOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnuz 12861 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
2 | 1zzd 12589 | . . . 4 ⊢ (⊤ → 1 ∈ ℤ) | |
3 | 0hmop 31214 | . . . . . . . 8 ⊢ 0hop ∈ HrmOp | |
4 | 3 | elexi 3494 | . . . . . . 7 ⊢ 0hop ∈ V |
5 | 4 | fvconst2 7200 | . . . . . 6 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0hop })‘𝑧) = 0hop ) |
6 | 5, 3 | eqeltrdi 2842 | . . . . 5 ⊢ (𝑧 ∈ ℕ → ((ℕ × { 0hop })‘𝑧) ∈ HrmOp) |
7 | 6 | adantl 483 | . . . 4 ⊢ ((⊤ ∧ 𝑧 ∈ ℕ) → ((ℕ × { 0hop })‘𝑧) ∈ HrmOp) |
8 | opsqrlem2.1 | . . . . . . 7 ⊢ 𝑇 ∈ HrmOp | |
9 | opsqrlem2.2 | . . . . . . 7 ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) | |
10 | opsqrlem2.3 | . . . . . . 7 ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) | |
11 | 8, 9, 10 | opsqrlem3 31373 | . . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) = (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))))) |
12 | halfre 12422 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ | |
13 | simpl 484 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → 𝑧 ∈ HrmOp) | |
14 | eqidd 2734 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧)) | |
15 | hmopco 31254 | . . . . . . . . . 10 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑧 ∈ HrmOp ∧ (𝑧 ∘ 𝑧) = (𝑧 ∘ 𝑧)) → (𝑧 ∘ 𝑧) ∈ HrmOp) | |
16 | 13, 13, 14, 15 | syl3anc 1372 | . . . . . . . . 9 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 ∘ 𝑧) ∈ HrmOp) |
17 | hmopd 31253 | . . . . . . . . 9 ⊢ ((𝑇 ∈ HrmOp ∧ (𝑧 ∘ 𝑧) ∈ HrmOp) → (𝑇 −op (𝑧 ∘ 𝑧)) ∈ HrmOp) | |
18 | 8, 16, 17 | sylancr 588 | . . . . . . . 8 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑇 −op (𝑧 ∘ 𝑧)) ∈ HrmOp) |
19 | hmopm 31252 | . . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ (𝑇 −op (𝑧 ∘ 𝑧)) ∈ HrmOp) → ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))) ∈ HrmOp) | |
20 | 12, 18, 19 | sylancr 588 | . . . . . . 7 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))) ∈ HrmOp) |
21 | hmops 31251 | . . . . . . 7 ⊢ ((𝑧 ∈ HrmOp ∧ ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧))) ∈ HrmOp) → (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))) ∈ HrmOp) | |
22 | 20, 21 | syldan 592 | . . . . . 6 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧 +op ((1 / 2) ·op (𝑇 −op (𝑧 ∘ 𝑧)))) ∈ HrmOp) |
23 | 11, 22 | eqeltrd 2834 | . . . . 5 ⊢ ((𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp) → (𝑧𝑆𝑤) ∈ HrmOp) |
24 | 23 | adantl 483 | . . . 4 ⊢ ((⊤ ∧ (𝑧 ∈ HrmOp ∧ 𝑤 ∈ HrmOp)) → (𝑧𝑆𝑤) ∈ HrmOp) |
25 | 1, 2, 7, 24 | seqf 13985 | . . 3 ⊢ (⊤ → seq1(𝑆, (ℕ × { 0hop })):ℕ⟶HrmOp) |
26 | 25 | mptru 1549 | . 2 ⊢ seq1(𝑆, (ℕ × { 0hop })):ℕ⟶HrmOp |
27 | 10 | feq1i 6705 | . 2 ⊢ (𝐹:ℕ⟶HrmOp ↔ seq1(𝑆, (ℕ × { 0hop })):ℕ⟶HrmOp) |
28 | 26, 27 | mpbir 230 | 1 ⊢ 𝐹:ℕ⟶HrmOp |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 {csn 4627 × cxp 5673 ∘ ccom 5679 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ∈ cmpo 7406 ℝcr 11105 1c1 11107 / cdiv 11867 ℕcn 12208 2c2 12263 seqcseq 13962 +op chos 30169 ·op chot 30170 −op chod 30171 0hop ch0o 30174 HrmOpcho 30181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30230 ax-hfvadd 30231 ax-hvcom 30232 ax-hvass 30233 ax-hv0cl 30234 ax-hvaddid 30235 ax-hfvmul 30236 ax-hvmulid 30237 ax-hvmulass 30238 ax-hvdistr1 30239 ax-hvdistr2 30240 ax-hvmul0 30241 ax-hfi 30310 ax-his1 30313 ax-his2 30314 ax-his3 30315 ax-his4 30316 ax-hcompl 30433 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7665 df-om 7851 df-1st 7970 df-2nd 7971 df-supp 8142 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19643 df-psmet 20921 df-xmet 20922 df-met 20923 df-bl 20924 df-mopn 20925 df-fbas 20926 df-fg 20927 df-cnfld 20930 df-top 22378 df-topon 22395 df-topsp 22417 df-bases 22431 df-cld 22505 df-ntr 22506 df-cls 22507 df-nei 22584 df-cn 22713 df-cnp 22714 df-lm 22715 df-haus 22801 df-tx 23048 df-hmeo 23241 df-fil 23332 df-fm 23424 df-flim 23425 df-flf 23426 df-xms 23808 df-ms 23809 df-tms 23810 df-cfil 24754 df-cau 24755 df-cmet 24756 df-grpo 29724 df-gid 29725 df-ginv 29726 df-gdiv 29727 df-ablo 29776 df-vc 29790 df-nv 29823 df-va 29826 df-ba 29827 df-sm 29828 df-0v 29829 df-vs 29830 df-nmcv 29831 df-ims 29832 df-dip 29932 df-ssp 29953 df-ph 30044 df-cbn 30094 df-hnorm 30199 df-hba 30200 df-hvsub 30202 df-hlim 30203 df-hcau 30204 df-sh 30438 df-ch 30452 df-oc 30483 df-ch0 30484 df-shs 30539 df-pjh 30626 df-hosum 30961 df-homul 30962 df-hodif 30963 df-h0op 30979 df-hmop 31075 |
This theorem is referenced by: opsqrlem5 31375 opsqrlem6 31376 |
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