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Mirrors > Home > MPE Home > Th. List > cphipcl | Structured version Visualization version GIF version |
Description: An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.) |
Ref | Expression |
---|---|
nmsq.v | β’ π = (Baseβπ) |
nmsq.h | β’ , = (Β·πβπ) |
Ref | Expression |
---|---|
cphipcl | β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
2 | eqid 2728 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
3 | 1, 2 | cphsubrg 25102 | . . . 4 β’ (π β βPreHil β (Baseβ(Scalarβπ)) β (SubRingββfld)) |
4 | cnfldbas 21277 | . . . . 5 β’ β = (Baseββfld) | |
5 | 4 | subrgss 20505 | . . . 4 β’ ((Baseβ(Scalarβπ)) β (SubRingββfld) β (Baseβ(Scalarβπ)) β β) |
6 | 3, 5 | syl 17 | . . 3 β’ (π β βPreHil β (Baseβ(Scalarβπ)) β β) |
7 | 6 | 3ad2ant1 1131 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (Baseβ(Scalarβπ)) β β) |
8 | cphphl 25093 | . . 3 β’ (π β βPreHil β π β PreHil) | |
9 | nmsq.h | . . . 4 β’ , = (Β·πβπ) | |
10 | nmsq.v | . . . 4 β’ π = (Baseβπ) | |
11 | 1, 9, 10, 2 | ipcl 21559 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (Baseβ(Scalarβπ))) |
12 | 8, 11 | syl3an1 1161 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (Baseβ(Scalarβπ))) |
13 | 7, 12 | sseldd 3980 | 1 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1085 = wceq 1534 β wcel 2099 β wss 3945 βcfv 6543 (class class class)co 7415 βcc 11131 Basecbs 17174 Scalarcsca 17230 Β·πcip 17232 SubRingcsubrg 20500 βfldccnfld 21273 PreHilcphl 21550 βPreHilccph 25088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-addf 11212 ax-mulf 11213 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-seq 13994 df-exp 14054 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-subg 19072 df-ghm 19162 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-cring 20170 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-subrg 20502 df-drng 20620 df-lmhm 20901 df-lvec 20982 df-sra 21052 df-rgmod 21053 df-cnfld 21274 df-phl 21552 df-cph 25090 |
This theorem is referenced by: nmsq 25116 cphipipcj 25122 cphassr 25134 cph2ass 25135 cphpyth 25138 cphipval2 25163 ipcnlem2 25166 pjthlem1 25359 |
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