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Mirrors > Home > MPE Home > Th. List > cphassir | Structured version Visualization version GIF version |
Description: "Associative" law for the second argument of an inner product with scalar _𝑖. (Contributed by AV, 17-Oct-2021.) |
Ref | Expression |
---|---|
cphassi.x | ⊢ 𝑋 = (Base‘𝑊) |
cphassi.s | ⊢ · = ( ·𝑠 ‘𝑊) |
cphassi.i | ⊢ , = (·𝑖‘𝑊) |
cphassi.f | ⊢ 𝐹 = (Scalar‘𝑊) |
cphassi.k | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
cphassir | ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1198 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ ℂPreHil) | |
2 | simp1r 1199 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ 𝐾) | |
3 | simp2 1138 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
4 | simp3 1139 | . . 3 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
5 | cphassi.i | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
6 | cphassi.x | . . . 4 ⊢ 𝑋 = (Base‘𝑊) | |
7 | cphassi.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
8 | cphassi.k | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
9 | cphassi.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
10 | 5, 6, 7, 8, 9 | cphassr 23967 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i ∈ 𝐾 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 , (i · 𝐵)) = ((∗‘i) · (𝐴 , 𝐵))) |
11 | 1, 2, 3, 4, 10 | syl13anc 1373 | . 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = ((∗‘i) · (𝐴 , 𝐵))) |
12 | cji 14611 | . . 3 ⊢ (∗‘i) = -i | |
13 | 12 | oveq1i 7183 | . 2 ⊢ ((∗‘i) · (𝐴 , 𝐵)) = (-i · (𝐴 , 𝐵)) |
14 | 11, 13 | eqtrdi 2790 | 1 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 ici 10620 · cmul 10623 -cneg 10952 ∗ccj 14548 Basecbs 16589 Scalarcsca 16674 ·𝑠 cvsca 16675 ·𝑖cip 16676 ℂPreHilccph 23921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-addf 10697 ax-mulf 10698 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-1st 7717 df-2nd 7718 df-tpos 7924 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-fz 12985 df-seq 13464 df-exp 13525 df-cj 14551 df-re 14552 df-im 14553 df-struct 16591 df-ndx 16592 df-slot 16593 df-base 16595 df-sets 16596 df-ress 16597 df-plusg 16684 df-mulr 16685 df-starv 16686 df-sca 16687 df-vsca 16688 df-ip 16689 df-tset 16690 df-ple 16691 df-ds 16693 df-unif 16694 df-0g 16821 df-mgm 17971 df-sgrp 18020 df-mnd 18031 df-mhm 18075 df-grp 18225 df-subg 18397 df-ghm 18477 df-cmn 19029 df-mgp 19362 df-ur 19374 df-ring 19421 df-cring 19422 df-oppr 19498 df-dvdsr 19516 df-unit 19517 df-rnghom 19592 df-drng 19626 df-subrg 19655 df-staf 19738 df-srng 19739 df-lmod 19758 df-lmhm 19916 df-lvec 19997 df-sra 20066 df-rgmod 20067 df-cnfld 20221 df-phl 20445 df-nlm 23342 df-clm 23818 df-cph 23923 |
This theorem is referenced by: cphipval2 23996 |
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