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| Mirrors > Home > MPE Home > Th. List > cph2ass | Structured version Visualization version GIF version | ||
| Description: Move scalar multiplication to outside of inner product. See his35 29351. (Contributed by Mario Carneiro, 17-Oct-2015.) |
| Ref | Expression |
|---|---|
| cphipcj.h | ⊢ , = (·𝑖‘𝑊) |
| cphipcj.v | ⊢ 𝑉 = (Base‘𝑊) |
| cphass.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| cphass.k | ⊢ 𝐾 = (Base‘𝐹) |
| cphass.s | ⊢ · = ( ·𝑠 ‘𝑊) |
| Ref | Expression |
|---|---|
| cph2ass | ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1 1134 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂPreHil) | |
| 2 | simp2r 1198 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝐾) | |
| 3 | simp3l 1199 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
| 4 | simp3r 1200 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐷 ∈ 𝑉) | |
| 5 | cphipcj.h | . . . . 5 ⊢ , = (·𝑖‘𝑊) | |
| 6 | cphipcj.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 7 | cphass.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 8 | cphass.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
| 9 | cphass.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
| 10 | 5, 6, 7, 8, 9 | cphassr 24281 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 11 | 1, 2, 3, 4, 10 | syl13anc 1370 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 12 | 11 | oveq2d 7271 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 · (𝐶 , (𝐵 · 𝐷))) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 13 | simp2l 1197 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
| 14 | cphlmod 24243 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | |
| 15 | 14 | 3ad2ant1 1131 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ LMod) |
| 16 | 6, 7, 9, 8 | lmodvscl 20055 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝐷 ∈ 𝑉) → (𝐵 · 𝐷) ∈ 𝑉) |
| 17 | 15, 2, 4, 16 | syl3anc 1369 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 · 𝐷) ∈ 𝑉) |
| 18 | 5, 6, 7, 8, 9 | cphass 24280 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ (𝐵 · 𝐷) ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 19 | 1, 13, 3, 17, 18 | syl13anc 1370 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 20 | cphclm 24258 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
| 21 | 20 | 3ad2ant1 1131 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂMod) |
| 22 | 7, 8 | clmsscn 24148 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ) |
| 23 | 21, 22 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐾 ⊆ ℂ) |
| 24 | 23, 13 | sseldd 3918 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ ℂ) |
| 25 | 23, 2 | sseldd 3918 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ ℂ) |
| 26 | 25 | cjcld 14835 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (∗‘𝐵) ∈ ℂ) |
| 27 | 6, 5 | cphipcl 24260 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 , 𝐷) ∈ ℂ) |
| 28 | 27 | 3expb 1118 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 29 | 28 | 3adant2 1129 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 30 | 24, 26, 29 | mulassd 10929 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷)) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 31 | 12, 19, 30 | 3eqtr4d 2788 | 1 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ⊆ wss 3883 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 · cmul 10807 ∗ccj 14735 Basecbs 16840 Scalarcsca 16891 ·𝑠 cvsca 16892 ·𝑖cip 16893 LModclmod 20038 ℂModcclm 24131 ℂPreHilccph 24235 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-addf 10881 ax-mulf 10882 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-grp 18495 df-subg 18667 df-ghm 18747 df-cmn 19303 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-rnghom 19874 df-drng 19908 df-subrg 19937 df-staf 20020 df-srng 20021 df-lmod 20040 df-lmhm 20199 df-lvec 20280 df-sra 20349 df-rgmod 20350 df-cnfld 20511 df-phl 20743 df-nlm 23648 df-clm 24132 df-cph 24237 |
| This theorem is referenced by: pjthlem1 24506 |
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