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| Mirrors > Home > MPE Home > Th. List > cphipcj | Structured version Visualization version GIF version | ||
| Description: Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj 21406. (Contributed by Mario Carneiro, 16-Oct-2015.) |
| Ref | Expression |
|---|---|
| cphipcj.h | β’ , = (Β·πβπ) |
| cphipcj.v | β’ π = (Baseβπ) |
| Ref | Expression |
|---|---|
| cphipcj | β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (ββ(π΄ , π΅)) = (π΅ , π΄)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cphclm 24930 | . . . . 5 β’ (π β βPreHil β π β βMod) | |
| 2 | eqid 2732 | . . . . . 6 β’ (Scalarβπ) = (Scalarβπ) | |
| 3 | 2 | clmcj 24816 | . . . . 5 β’ (π β βMod β β = (*πβ(Scalarβπ))) |
| 4 | 1, 3 | syl 17 | . . . 4 β’ (π β βPreHil β β = (*πβ(Scalarβπ))) |
| 5 | 4 | 3ad2ant1 1133 | . . 3 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β β = (*πβ(Scalarβπ))) |
| 6 | 5 | fveq1d 6893 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (ββ(π΄ , π΅)) = ((*πβ(Scalarβπ))β(π΄ , π΅))) |
| 7 | cphphl 24912 | . . 3 β’ (π β βPreHil β π β PreHil) | |
| 8 | cphipcj.h | . . . 4 β’ , = (Β·πβπ) | |
| 9 | cphipcj.v | . . . 4 β’ π = (Baseβπ) | |
| 10 | eqid 2732 | . . . 4 β’ (*πβ(Scalarβπ)) = (*πβ(Scalarβπ)) | |
| 11 | 2, 8, 9, 10 | ipcj 21406 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β ((*πβ(Scalarβπ))β(π΄ , π΅)) = (π΅ , π΄)) |
| 12 | 7, 11 | syl3an1 1163 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β ((*πβ(Scalarβπ))β(π΄ , π΅)) = (π΅ , π΄)) |
| 13 | 6, 12 | eqtrd 2772 | 1 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (ββ(π΄ , π΅)) = (π΅ , π΄)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1087 = wceq 1541 β wcel 2106 βcfv 6543 (class class class)co 7411 βccj 15047 Basecbs 17148 *πcstv 17203 Scalarcsca 17204 Β·πcip 17206 PreHilcphl 21396 βModcclm 24802 βPreHilccph 24907 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-addf 11191 ax-mulf 11192 |
| This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-seq 13971 df-exp 14032 df-cj 15050 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-subrg 20459 df-drng 20502 df-lvec 20858 df-cnfld 21145 df-phl 21398 df-nlm 24315 df-clm 24803 df-cph 24909 |
| This theorem is referenced by: cphipipcj 24941 pjthlem1 25178 |
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