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Mirrors > Home > MPE Home > Th. List > tcphcphlem3 | Structured version Visualization version GIF version |
Description: Lemma for tcphcph 24624: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.) |
Ref | Expression |
---|---|
tcphval.n | β’ πΊ = (toβPreHilβπ) |
tcphcph.v | β’ π = (Baseβπ) |
tcphcph.f | β’ πΉ = (Scalarβπ) |
tcphcph.1 | β’ (π β π β PreHil) |
tcphcph.2 | β’ (π β πΉ = (βfld βΎs πΎ)) |
tcphcph.h | β’ , = (Β·πβπ) |
Ref | Expression |
---|---|
tcphcphlem3 | β’ ((π β§ π β π) β (π , π) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tcphval.n | . . . . . 6 β’ πΊ = (toβPreHilβπ) | |
2 | tcphcph.v | . . . . . 6 β’ π = (Baseβπ) | |
3 | tcphcph.f | . . . . . 6 β’ πΉ = (Scalarβπ) | |
4 | tcphcph.1 | . . . . . 6 β’ (π β π β PreHil) | |
5 | tcphcph.2 | . . . . . 6 β’ (π β πΉ = (βfld βΎs πΎ)) | |
6 | 1, 2, 3, 4, 5 | phclm 24619 | . . . . 5 β’ (π β π β βMod) |
7 | 6 | adantr 482 | . . . 4 β’ ((π β§ π β π) β π β βMod) |
8 | eqid 2733 | . . . . 5 β’ (BaseβπΉ) = (BaseβπΉ) | |
9 | 3, 8 | clmsscn 24465 | . . . 4 β’ (π β βMod β (BaseβπΉ) β β) |
10 | 7, 9 | syl 17 | . . 3 β’ ((π β§ π β π) β (BaseβπΉ) β β) |
11 | tcphcph.h | . . . . . 6 β’ , = (Β·πβπ) | |
12 | 3, 11, 2, 8 | ipcl 21060 | . . . . 5 β’ ((π β PreHil β§ π β π β§ π β π) β (π , π) β (BaseβπΉ)) |
13 | 12 | 3anidm23 1422 | . . . 4 β’ ((π β PreHil β§ π β π) β (π , π) β (BaseβπΉ)) |
14 | 4, 13 | sylan 581 | . . 3 β’ ((π β§ π β π) β (π , π) β (BaseβπΉ)) |
15 | 10, 14 | sseldd 3949 | . 2 β’ ((π β§ π β π) β (π , π) β β) |
16 | 3 | clmcj 24462 | . . . . 5 β’ (π β βMod β β = (*πβπΉ)) |
17 | 7, 16 | syl 17 | . . . 4 β’ ((π β§ π β π) β β = (*πβπΉ)) |
18 | 17 | fveq1d 6848 | . . 3 β’ ((π β§ π β π) β (ββ(π , π)) = ((*πβπΉ)β(π , π))) |
19 | 4 | adantr 482 | . . . 4 β’ ((π β§ π β π) β π β PreHil) |
20 | simpr 486 | . . . 4 β’ ((π β§ π β π) β π β π) | |
21 | eqid 2733 | . . . . 5 β’ (*πβπΉ) = (*πβπΉ) | |
22 | 3, 11, 2, 21 | ipcj 21061 | . . . 4 β’ ((π β PreHil β§ π β π β§ π β π) β ((*πβπΉ)β(π , π)) = (π , π)) |
23 | 19, 20, 20, 22 | syl3anc 1372 | . . 3 β’ ((π β§ π β π) β ((*πβπΉ)β(π , π)) = (π , π)) |
24 | 18, 23 | eqtrd 2773 | . 2 β’ ((π β§ π β π) β (ββ(π , π)) = (π , π)) |
25 | 15, 24 | cjrebd 15096 | 1 β’ ((π β§ π β π) β (π , π) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3914 βcfv 6500 (class class class)co 7361 βcc 11057 βcr 11058 βccj 14990 Basecbs 17091 βΎs cress 17120 *πcstv 17143 Scalarcsca 17144 Β·πcip 17146 βfldccnfld 20819 PreHilcphl 21051 βModcclm 24448 toβPreHilctcph 24554 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-addf 11138 ax-mulf 11139 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-subg 18933 df-ghm 19014 df-cmn 19572 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-drng 20221 df-subrg 20262 df-lmhm 20527 df-lvec 20608 df-sra 20678 df-rgmod 20679 df-cnfld 20820 df-phl 21053 df-clm 24449 |
This theorem is referenced by: ipcau2 24621 tcphcphlem1 24622 tcphcphlem2 24623 tcphcph 24624 |
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