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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 29123. (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 1138 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂPreHil) | |
| 2 | simp2r 1202 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝐾) | |
| 3 | simp3l 1203 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
| 4 | simp3r 1204 | . . . 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 24063 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 11 | 1, 2, 3, 4, 10 | syl13anc 1374 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 12 | 11 | oveq2d 7207 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 · (𝐶 , (𝐵 · 𝐷))) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 13 | simp2l 1201 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
| 14 | cphlmod 24025 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | |
| 15 | 14 | 3ad2ant1 1135 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ LMod) |
| 16 | 6, 7, 9, 8 | lmodvscl 19870 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝐷 ∈ 𝑉) → (𝐵 · 𝐷) ∈ 𝑉) |
| 17 | 15, 2, 4, 16 | syl3anc 1373 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 · 𝐷) ∈ 𝑉) |
| 18 | 5, 6, 7, 8, 9 | cphass 24062 | . . 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 1374 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 20 | cphclm 24040 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
| 21 | 20 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂMod) |
| 22 | 7, 8 | clmsscn 23930 | . . . . 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 3888 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ ℂ) |
| 25 | 23, 2 | sseldd 3888 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ ℂ) |
| 26 | 25 | cjcld 14724 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (∗‘𝐵) ∈ ℂ) |
| 27 | 6, 5 | cphipcl 24042 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 , 𝐷) ∈ ℂ) |
| 28 | 27 | 3expb 1122 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 29 | 28 | 3adant2 1133 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 30 | 24, 26, 29 | mulassd 10821 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷)) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 31 | 12, 19, 30 | 3eqtr4d 2781 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ⊆ wss 3853 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 · cmul 10699 ∗ccj 14624 Basecbs 16666 Scalarcsca 16752 ·𝑠 cvsca 16753 ·𝑖cip 16754 LModclmod 19853 ℂModcclm 23913 ℂPreHilccph 24017 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774 |
| This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-seq 13540 df-exp 13601 df-cj 14627 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-grp 18322 df-subg 18494 df-ghm 18574 df-cmn 19126 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-rnghom 19689 df-drng 19723 df-subrg 19752 df-staf 19835 df-srng 19836 df-lmod 19855 df-lmhm 20013 df-lvec 20094 df-sra 20163 df-rgmod 20164 df-cnfld 20318 df-phl 20542 df-nlm 23438 df-clm 23914 df-cph 24019 |
| This theorem is referenced by: pjthlem1 24288 |
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