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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 28275. (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 1131 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂPreHil) | |
| 2 | simp2r 1243 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ 𝐾) | |
| 3 | simp3l 1244 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
| 4 | simp3r 1245 | . . . 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 23232 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 11 | 1, 2, 3, 4, 10 | syl13anc 1479 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , (𝐵 · 𝐷)) = ((∗‘𝐵) · (𝐶 , 𝐷))) |
| 12 | 11 | oveq2d 6830 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 · (𝐶 , (𝐵 · 𝐷))) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 13 | simp2l 1242 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
| 14 | cphlmod 23194 | . . . . 5 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod) | |
| 15 | 14 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ LMod) |
| 16 | 6, 7, 9, 8 | lmodvscl 19102 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐵 ∈ 𝐾 ∧ 𝐷 ∈ 𝑉) → (𝐵 · 𝐷) ∈ 𝑉) |
| 17 | 15, 2, 4, 16 | syl3anc 1477 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 · 𝐷) ∈ 𝑉) |
| 18 | 5, 6, 7, 8, 9 | cphass 23231 | . . 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 1479 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = (𝐴 · (𝐶 , (𝐵 · 𝐷)))) |
| 20 | cphclm 23209 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
| 21 | 20 | 3ad2ant1 1128 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝑊 ∈ ℂMod) |
| 22 | 7, 8 | clmsscn 23099 | . . . . 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 3745 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐴 ∈ ℂ) |
| 25 | 23, 2 | sseldd 3745 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → 𝐵 ∈ ℂ) |
| 26 | 25 | cjcld 14155 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (∗‘𝐵) ∈ ℂ) |
| 27 | 6, 5 | cphipcl 23211 | . . . . 5 ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 , 𝐷) ∈ ℂ) |
| 28 | 27 | 3expb 1114 | . . . 4 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 29 | 28 | 3adant2 1126 | . . 3 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐶 , 𝐷) ∈ ℂ) |
| 30 | 24, 26, 29 | mulassd 10275 | . 2 ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷)) = (𝐴 · ((∗‘𝐵) · (𝐶 , 𝐷)))) |
| 31 | 12, 19, 30 | 3eqtr4d 2804 | 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 1072 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 ‘cfv 6049 (class class class)co 6814 ℂcc 10146 · cmul 10153 ∗ccj 14055 Basecbs 16079 Scalarcsca 16166 ·𝑠 cvsca 16167 ·𝑖cip 16168 LModclmod 19085 ℂModcclm 23082 ℂPreHilccph 23186 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-addf 10227 ax-mulf 10228 |
| This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-div 10897 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-7 11296 df-8 11297 df-9 11298 df-n0 11505 df-z 11590 df-dec 11706 df-uz 11900 df-fz 12540 df-seq 13016 df-exp 13075 df-cj 14058 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-starv 16178 df-sca 16179 df-vsca 16180 df-ip 16181 df-tset 16182 df-ple 16183 df-ds 16186 df-unif 16187 df-0g 16324 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-mhm 17556 df-grp 17646 df-subg 17812 df-ghm 17879 df-cmn 18415 df-mgp 18710 df-ur 18722 df-ring 18769 df-cring 18770 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-rnghom 18937 df-drng 18971 df-subrg 19000 df-staf 19067 df-srng 19068 df-lmod 19087 df-lmhm 19244 df-lvec 19325 df-sra 19394 df-rgmod 19395 df-cnfld 19969 df-phl 20193 df-nlm 22612 df-clm 23083 df-cph 23188 |
| This theorem is referenced by: pjthlem1 23428 |
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