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| Mirrors > Home > MPE Home > Th. List > hlipcj | Structured version Visualization version GIF version | ||
| Description: Conjugate law for Hilbert space inner product. (Contributed by NM, 8-Sep-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hlipf.1 | β’ π = (BaseSetβπ) |
| hlipf.7 | β’ π = (Β·πOLDβπ) |
| Ref | Expression |
|---|---|
| hlipcj | β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (ββ(π΅ππ΄))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlnv 30144 | . . . 4 β’ (π β CHilOLD β π β NrmCVec) | |
| 2 | hlipf.1 | . . . . 5 β’ π = (BaseSetβπ) | |
| 3 | hlipf.7 | . . . . 5 β’ π = (Β·πOLDβπ) | |
| 4 | 2, 3 | dipcj 29967 | . . . 4 β’ ((π β NrmCVec β§ π΅ β π β§ π΄ β π) β (ββ(π΅ππ΄)) = (π΄ππ΅)) |
| 5 | 1, 4 | syl3an1 1164 | . . 3 β’ ((π β CHilOLD β§ π΅ β π β§ π΄ β π) β (ββ(π΅ππ΄)) = (π΄ππ΅)) |
| 6 | 5 | 3com23 1127 | . 2 β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (ββ(π΅ππ΄)) = (π΄ππ΅)) |
| 7 | 6 | eqcomd 2739 | 1 β’ ((π β CHilOLD β§ π΄ β π β§ π΅ β π) β (π΄ππ΅) = (ββ(π΅ππ΄))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1088 = wceq 1542 β wcel 2107 βcfv 6544 (class class class)co 7409 βccj 15043 NrmCVeccnv 29837 BaseSetcba 29839 Β·πOLDcdip 29953 CHilOLDchlo 30138 |
| 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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
| 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 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-grpo 29746 df-gid 29747 df-ginv 29748 df-ablo 29798 df-vc 29812 df-nv 29845 df-va 29848 df-ba 29849 df-sm 29850 df-0v 29851 df-nmcv 29853 df-dip 29954 df-cbn 30116 df-hlo 30139 |
| This theorem is referenced by: axhis1-zf 30247 |
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