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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 2724 | . . . . 5 β’ (Scalarβπ) = (Scalarβπ) | |
2 | eqid 2724 | . . . . 5 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
3 | 1, 2 | cphsubrg 25032 | . . . 4 β’ (π β βPreHil β (Baseβ(Scalarβπ)) β (SubRingββfld)) |
4 | cnfldbas 21234 | . . . . 5 β’ β = (Baseββfld) | |
5 | 4 | subrgss 20466 | . . . 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 1130 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (Baseβ(Scalarβπ)) β β) |
8 | cphphl 25023 | . . 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 21496 | . . 3 β’ ((π β PreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (Baseβ(Scalarβπ))) |
12 | 8, 11 | syl3an1 1160 | . 2 β’ ((π β βPreHil β§ π΄ β π β§ π΅ β π) β (π΄ , π΅) β (Baseβ(Scalarβπ))) |
13 | 7, 12 | sseldd 3976 | 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 1084 = wceq 1533 β wcel 2098 β wss 3941 βcfv 6534 (class class class)co 7402 βcc 11105 Basecbs 17145 Scalarcsca 17201 Β·πcip 17203 SubRingcsubrg 20461 βfldccnfld 21230 PreHilcphl 21487 βPreHilccph 25018 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-8 12279 df-9 12280 df-n0 12471 df-z 12557 df-dec 12676 df-uz 12821 df-fz 13483 df-seq 13965 df-exp 14026 df-struct 17081 df-sets 17098 df-slot 17116 df-ndx 17128 df-base 17146 df-ress 17175 df-plusg 17211 df-mulr 17212 df-starv 17213 df-sca 17214 df-vsca 17215 df-ip 17216 df-tset 17217 df-ple 17218 df-ds 17220 df-unif 17221 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-subg 19042 df-ghm 19131 df-cmn 19694 df-abl 19695 df-mgp 20032 df-rng 20050 df-ur 20079 df-ring 20132 df-cring 20133 df-oppr 20228 df-dvdsr 20251 df-unit 20252 df-subrg 20463 df-drng 20581 df-lmhm 20862 df-lvec 20943 df-sra 21013 df-rgmod 21014 df-cnfld 21231 df-phl 21489 df-cph 25020 |
This theorem is referenced by: nmsq 25046 cphipipcj 25052 cphassr 25064 cph2ass 25065 cphpyth 25068 cphipval2 25093 ipcnlem2 25096 pjthlem1 25289 |
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