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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 2732 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
2 | eqid 2732 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
3 | 1, 2 | cphsubrg 24696 | . . . 4 β’ (π β βPreHil β (Baseβ(Scalarβπ)) β (SubRingββfld)) |
4 | cnfldbas 20947 | . . . . 5 β’ β = (Baseββfld) | |
5 | 4 | subrgss 20319 | . . . 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 1133 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (Baseβ(Scalarβπ)) β β) |
8 | cphphl 24687 | . . 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 21185 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (Baseβ(Scalarβπ))) |
12 | 8, 11 | syl3an1 1163 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (Baseβ(Scalarβπ))) |
13 | 7, 12 | sseldd 3983 | 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 1087 = wceq 1541 β wcel 2106 β wss 3948 βcfv 6543 (class class class)co 7408 βcc 11107 Basecbs 17143 Scalarcsca 17199 Β·πcip 17201 SubRingcsubrg 20314 βfldccnfld 20943 PreHilcphl 21176 βPreHilccph 24682 |
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 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188 ax-mulf 11189 |
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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-1st 7974 df-2nd 7975 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13484 df-seq 13966 df-exp 14027 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17386 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-grp 18821 df-subg 19002 df-ghm 19089 df-cmn 19649 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-subrg 20316 df-drng 20358 df-lmhm 20632 df-lvec 20713 df-sra 20784 df-rgmod 20785 df-cnfld 20944 df-phl 21178 df-cph 24684 |
This theorem is referenced by: nmsq 24710 cphipipcj 24716 cphassr 24728 cph2ass 24729 cphpyth 24732 cphipval2 24757 ipcnlem2 24760 pjthlem1 24953 |
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