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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 27123. (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 1054 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂPreHil) | |
| 2 | simp2r 1081 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝐾) | |
| 3 | simp3l 1082 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
| 4 | simp3r 1083 | . . . 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 22765 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 11 | 1, 2, 3, 4, 10 | syl13anc 1320 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 12 | 11 | oveq2d 6543 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 · (𝐶 , (𝐵 · 𝐷))) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 13 | simp2l 1080 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
| 14 | cphlmod 22727 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | |
| 15 | 14 | 3ad2ant1 1075 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ LMod) |
| 16 | 6, 7, 9, 8 | lmodvscl 18652 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝐷 ∈ 𝑉) → (𝐵 · 𝐷) ∈ 𝑉) |
| 17 | 15, 2, 4, 16 | syl3anc 1318 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 · 𝐷) ∈ 𝑉) |
| 18 | 5, 6, 7, 8, 9 | cphass 22764 | . . 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 1320 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 20 | cphclm 22742 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
| 21 | 20 | 3ad2ant1 1075 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂMod) |
| 22 | 7, 8 | clmsscn 22635 | . . . . 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 3569 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ ℂ) |
| 25 | 23, 2 | sseldd 3569 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ ℂ) |
| 26 | 25 | cjcld 13733 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (∗‘𝐵) ∈ ℂ) |
| 27 | 6, 5 | cphipcl 22744 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 , 𝐷) ∈ ℂ) |
| 28 | 27 | 3expb 1258 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 29 | 28 | 3adant2 1073 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 30 | 24, 26, 29 | mulassd 9920 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷)) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 31 | 12, 19, 30 | 3eqtr4d 2654 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5790 (class class class)co 6527 ℂcc 9791 · cmul 9798 ∗ccj 13633 Basecbs 15644 Scalarcsca 15720 ·𝑠 cvsca 15721 ·𝑖cip 15722 LModclmod 18635 ℂModcclm 22618 ℂPreHilccph 22719 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4694 ax-sep 4704 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825 ax-cnex 9849 ax-resscn 9850 ax-1cn 9851 ax-icn 9852 ax-addcl 9853 ax-addrcl 9854 ax-mulcl 9855 ax-mulrcl 9856 ax-mulcom 9857 ax-addass 9858 ax-mulass 9859 ax-distr 9860 ax-i2m1 9861 ax-1ne0 9862 ax-1rid 9863 ax-rnegex 9864 ax-rrecex 9865 ax-cnre 9866 ax-pre-lttri 9867 ax-pre-lttrn 9868 ax-pre-ltadd 9869 ax-pre-mulgt0 9870 ax-addf 9872 ax-mulf 9873 |
| This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4368 df-int 4406 df-iun 4452 df-br 4579 df-opab 4639 df-mpt 4640 df-tr 4676 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-om 6936 df-1st 7037 df-2nd 7038 df-tpos 7217 df-wrecs 7272 df-recs 7333 df-rdg 7371 df-1o 7425 df-oadd 7429 df-er 7607 df-map 7724 df-en 7820 df-dom 7821 df-sdom 7822 df-fin 7823 df-pnf 9933 df-mnf 9934 df-xr 9935 df-ltxr 9936 df-le 9937 df-sub 10120 df-neg 10121 df-div 10537 df-nn 10871 df-2 10929 df-3 10930 df-4 10931 df-5 10932 df-6 10933 df-7 10934 df-8 10935 df-9 10936 df-n0 11143 df-z 11214 df-dec 11329 df-uz 11523 df-fz 12156 df-seq 12622 df-exp 12681 df-cj 13636 df-struct 15646 df-ndx 15647 df-slot 15648 df-base 15649 df-sets 15650 df-ress 15651 df-plusg 15730 df-mulr 15731 df-starv 15732 df-sca 15733 df-vsca 15734 df-ip 15735 df-tset 15736 df-ple 15737 df-ds 15740 df-unif 15741 df-0g 15874 df-mgm 17014 df-sgrp 17056 df-mnd 17067 df-mhm 17107 df-grp 17197 df-subg 17363 df-ghm 17430 df-cmn 17967 df-mgp 18262 df-ur 18274 df-ring 18321 df-cring 18322 df-oppr 18395 df-dvdsr 18413 df-unit 18414 df-rnghom 18487 df-drng 18521 df-subrg 18550 df-staf 18617 df-srng 18618 df-lmod 18637 df-lmhm 18792 df-lvec 18873 df-sra 18942 df-rgmod 18943 df-cnfld 19517 df-phl 19738 df-nlm 22149 df-clm 22619 df-cph 22721 |
| This theorem is referenced by: pjthlem1 22961 |
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