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Mirrors > Home > HSE Home > Th. List > opsqrlem5 | 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 |
---|---|
opsqrlem5 | ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnnuz 12620 | . . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
2 | seqp1 13734 | . . . 4 ⊢ (𝑁 ∈ (ℤ≥‘1) → (seq1(𝑆, (ℕ × { 0hop }))‘(𝑁 + 1)) = ((seq1(𝑆, (ℕ × { 0hop }))‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1)))) | |
3 | 1, 2 | sylbi 216 | . . 3 ⊢ (𝑁 ∈ ℕ → (seq1(𝑆, (ℕ × { 0hop }))‘(𝑁 + 1)) = ((seq1(𝑆, (ℕ × { 0hop }))‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1)))) |
4 | opsqrlem2.3 | . . . 4 ⊢ 𝐹 = seq1(𝑆, (ℕ × { 0hop })) | |
5 | 4 | fveq1i 6777 | . . 3 ⊢ (𝐹‘(𝑁 + 1)) = (seq1(𝑆, (ℕ × { 0hop }))‘(𝑁 + 1)) |
6 | 4 | fveq1i 6777 | . . . 4 ⊢ (𝐹‘𝑁) = (seq1(𝑆, (ℕ × { 0hop }))‘𝑁) |
7 | 6 | oveq1i 7287 | . . 3 ⊢ ((𝐹‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1))) = ((seq1(𝑆, (ℕ × { 0hop }))‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1))) |
8 | 3, 5, 7 | 3eqtr4g 2803 | . 2 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1)))) |
9 | opsqrlem2.1 | . . . . 5 ⊢ 𝑇 ∈ HrmOp | |
10 | opsqrlem2.2 | . . . . 5 ⊢ 𝑆 = (𝑥 ∈ HrmOp, 𝑦 ∈ HrmOp ↦ (𝑥 +op ((1 / 2) ·op (𝑇 −op (𝑥 ∘ 𝑥))))) | |
11 | 9, 10, 4 | opsqrlem4 30502 | . . . 4 ⊢ 𝐹:ℕ⟶HrmOp |
12 | 11 | ffvelrni 6962 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐹‘𝑁) ∈ HrmOp) |
13 | peano2nn 11983 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (𝑁 + 1) ∈ ℕ) | |
14 | 0hmop 30342 | . . . . . . 7 ⊢ 0hop ∈ HrmOp | |
15 | 14 | elexi 3450 | . . . . . 6 ⊢ 0hop ∈ V |
16 | 15 | fvconst2 7081 | . . . . 5 ⊢ ((𝑁 + 1) ∈ ℕ → ((ℕ × { 0hop })‘(𝑁 + 1)) = 0hop ) |
17 | 13, 16 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((ℕ × { 0hop })‘(𝑁 + 1)) = 0hop ) |
18 | 17, 14 | eqeltrdi 2847 | . . 3 ⊢ (𝑁 ∈ ℕ → ((ℕ × { 0hop })‘(𝑁 + 1)) ∈ HrmOp) |
19 | 9, 10, 4 | opsqrlem3 30501 | . . 3 ⊢ (((𝐹‘𝑁) ∈ HrmOp ∧ ((ℕ × { 0hop })‘(𝑁 + 1)) ∈ HrmOp) → ((𝐹‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1))) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁)))))) |
20 | 12, 18, 19 | syl2anc 584 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝐹‘𝑁)𝑆((ℕ × { 0hop })‘(𝑁 + 1))) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁)))))) |
21 | 8, 20 | eqtrd 2778 | 1 ⊢ (𝑁 ∈ ℕ → (𝐹‘(𝑁 + 1)) = ((𝐹‘𝑁) +op ((1 / 2) ·op (𝑇 −op ((𝐹‘𝑁) ∘ (𝐹‘𝑁)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 {csn 4563 × cxp 5589 ∘ ccom 5595 ‘cfv 6435 (class class class)co 7277 ∈ cmpo 7279 1c1 10870 + caddc 10872 / cdiv 11630 ℕcn 11971 2c2 12026 ℤ≥cuz 12580 seqcseq 13719 +op chos 29297 ·op chot 29298 −op chod 29299 0hop ch0o 29302 HrmOpcho 29309 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-inf2 9397 ax-cc 10189 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 ax-pre-sup 10947 ax-addf 10948 ax-mulf 10949 ax-hilex 29358 ax-hfvadd 29359 ax-hvcom 29360 ax-hvass 29361 ax-hv0cl 29362 ax-hvaddid 29363 ax-hfvmul 29364 ax-hvmulid 29365 ax-hvmulass 29366 ax-hvdistr1 29367 ax-hvdistr2 29368 ax-hvmul0 29369 ax-hfi 29438 ax-his1 29441 ax-his2 29442 ax-his3 29443 ax-his4 29444 ax-hcompl 29561 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-se 5547 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-isom 6444 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7976 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-1o 8295 df-2o 8296 df-oadd 8299 df-omul 8300 df-er 8496 df-map 8615 df-pm 8616 df-ixp 8684 df-en 8732 df-dom 8733 df-sdom 8734 df-fin 8735 df-fsupp 9127 df-fi 9168 df-sup 9199 df-inf 9200 df-oi 9267 df-card 9695 df-acn 9698 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-uz 12581 df-q 12687 df-rp 12729 df-xneg 12846 df-xadd 12847 df-xmul 12848 df-ioo 13081 df-ico 13083 df-icc 13084 df-fz 13238 df-fzo 13381 df-fl 13510 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-clim 15195 df-rlim 15196 df-sum 15396 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-fbas 20592 df-fg 20593 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-ntr 22169 df-cls 22170 df-nei 22247 df-cn 22376 df-cnp 22377 df-lm 22378 df-haus 22464 df-tx 22711 df-hmeo 22904 df-fil 22995 df-fm 23087 df-flim 23088 df-flf 23089 df-xms 23471 df-ms 23472 df-tms 23473 df-cfil 24417 df-cau 24418 df-cmet 24419 df-grpo 28852 df-gid 28853 df-ginv 28854 df-gdiv 28855 df-ablo 28904 df-vc 28918 df-nv 28951 df-va 28954 df-ba 28955 df-sm 28956 df-0v 28957 df-vs 28958 df-nmcv 28959 df-ims 28960 df-dip 29060 df-ssp 29081 df-ph 29172 df-cbn 29222 df-hnorm 29327 df-hba 29328 df-hvsub 29330 df-hlim 29331 df-hcau 29332 df-sh 29566 df-ch 29580 df-oc 29611 df-ch0 29612 df-shs 29667 df-pjh 29754 df-hosum 30089 df-homul 30090 df-hodif 30091 df-h0op 30107 df-hmop 30203 |
This theorem is referenced by: opsqrlem6 30504 |
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