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Mirrors > Home > MPE Home > Th. List > cphassi | Structured version Visualization version GIF version |
Description: Associative law for the first 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 |
---|---|
cphassi | ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1l 1211 | . 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝑊 ∈ ℂPreHil) | |
2 | simp1r 1212 | . 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → i ∈ 𝐾) | |
3 | simp3 1129 | . 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋) | |
4 | simp2 1128 | . 2 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
5 | cphassi.i | . . 3 ⊢ , = (·𝑖‘𝑊) | |
6 | cphassi.x | . . 3 ⊢ 𝑋 = (Base‘𝑊) | |
7 | cphassi.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
8 | cphassi.k | . . 3 ⊢ 𝐾 = (Base‘𝐹) | |
9 | cphassi.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
10 | 5, 6, 7, 8, 9 | cphass 23422 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴))) |
11 | 1, 2, 3, 4, 10 | syl13anc 1440 | 1 ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 ici 10276 · cmul 10279 Basecbs 16259 Scalarcsca 16345 ·𝑠 cvsca 16346 ·𝑖cip 16347 ℂPreHilccph 23377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-fz 12648 df-seq 13124 df-exp 13183 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-vsca 16359 df-ip 16360 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-subg 17979 df-cmn 18585 df-mgp 18881 df-ur 18893 df-ring 18940 df-cring 18941 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-drng 19145 df-subrg 19174 df-lmod 19261 df-lmhm 19421 df-lvec 19502 df-sra 19573 df-rgmod 19574 df-cnfld 20147 df-phl 20373 df-nlm 22803 df-clm 23274 df-cph 23379 |
This theorem is referenced by: cphipval2 23451 |
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