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Mirrors > Home > MPE Home > Th. List > cph2subdi | Structured version Visualization version GIF version |
Description: Distributive law for inner product subtraction. Complex version of ip2subdi 21633. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
cphipcj.h | ⊢ , = (·𝑖‘𝑊) |
cphipcj.v | ⊢ 𝑉 = (Base‘𝑊) |
cphsubdir.m | ⊢ − = (-g‘𝑊) |
cph2subdi.1 | ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) |
cph2subdi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
cph2subdi.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
cph2subdi.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
cph2subdi.5 | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
Ref | Expression |
---|---|
cph2subdi | ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷)) − ((𝐴 , 𝐷) + (𝐵 , 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cph2subdi.1 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) | |
2 | cphclm 25202 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
4 | eqid 2726 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | 4 | clmadd 25086 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → + = (+g‘(Scalar‘𝑊))) |
6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → + = (+g‘(Scalar‘𝑊))) |
7 | 6 | oveqd 7430 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐶) + (𝐵 , 𝐷)) = ((𝐴 , 𝐶)(+g‘(Scalar‘𝑊))(𝐵 , 𝐷))) |
8 | 6 | oveqd 7430 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) = ((𝐴 , 𝐷)(+g‘(Scalar‘𝑊))(𝐵 , 𝐶))) |
9 | 7, 8 | oveq12d 7431 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) + (𝐵 , 𝐷))(-g‘(Scalar‘𝑊))((𝐴 , 𝐷) + (𝐵 , 𝐶))) = (((𝐴 , 𝐶)(+g‘(Scalar‘𝑊))(𝐵 , 𝐷))(-g‘(Scalar‘𝑊))((𝐴 , 𝐷)(+g‘(Scalar‘𝑊))(𝐵 , 𝐶)))) |
10 | cphphl 25184 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
11 | 1, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ PreHil) |
12 | cph2subdi.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | cph2subdi.4 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
14 | cphipcj.h | . . . . . 6 ⊢ , = (·𝑖‘𝑊) | |
15 | cphipcj.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
16 | eqid 2726 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
17 | 4, 14, 15, 16 | ipcl 21622 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
18 | 11, 12, 13, 17 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
19 | cph2subdi.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
20 | cph2subdi.5 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
21 | 4, 14, 15, 16 | ipcl 21622 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
22 | 11, 19, 20, 21 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
23 | 4, 16 | clmacl 25096 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 , 𝐶) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝐵 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) → ((𝐴 , 𝐶) + (𝐵 , 𝐷)) ∈ (Base‘(Scalar‘𝑊))) |
24 | 3, 18, 22, 23 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐶) + (𝐵 , 𝐷)) ∈ (Base‘(Scalar‘𝑊))) |
25 | 4, 14, 15, 16 | ipcl 21622 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
26 | 11, 12, 20, 25 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
27 | 4, 14, 15, 16 | ipcl 21622 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
28 | 11, 19, 13, 27 | syl3anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
29 | 4, 16 | clmacl 25096 | . . . 4 ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 , 𝐷) ∈ (Base‘(Scalar‘𝑊)) ∧ (𝐵 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘(Scalar‘𝑊))) |
30 | 3, 26, 28, 29 | syl3anc 1368 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘(Scalar‘𝑊))) |
31 | 4, 16 | clmsub 25092 | . . 3 ⊢ ((𝑊 ∈ ℂMod ∧ ((𝐴 , 𝐶) + (𝐵 , 𝐷)) ∈ (Base‘(Scalar‘𝑊)) ∧ ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘(Scalar‘𝑊))) → (((𝐴 , 𝐶) + (𝐵 , 𝐷)) − ((𝐴 , 𝐷) + (𝐵 , 𝐶))) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))(-g‘(Scalar‘𝑊))((𝐴 , 𝐷) + (𝐵 , 𝐶)))) |
32 | 3, 24, 30, 31 | syl3anc 1368 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) + (𝐵 , 𝐷)) − ((𝐴 , 𝐷) + (𝐵 , 𝐶))) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))(-g‘(Scalar‘𝑊))((𝐴 , 𝐷) + (𝐵 , 𝐶)))) |
33 | cphsubdir.m | . . 3 ⊢ − = (-g‘𝑊) | |
34 | eqid 2726 | . . 3 ⊢ (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑊)) | |
35 | eqid 2726 | . . 3 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
36 | 4, 14, 15, 33, 34, 35, 11, 12, 19, 13, 20 | ip2subdi 21633 | . 2 ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶)(+g‘(Scalar‘𝑊))(𝐵 , 𝐷))(-g‘(Scalar‘𝑊))((𝐴 , 𝐷)(+g‘(Scalar‘𝑊))(𝐵 , 𝐶)))) |
37 | 9, 32, 36 | 3eqtr4rd 2777 | 1 ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷)) − ((𝐴 , 𝐷) + (𝐵 , 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6543 (class class class)co 7413 + caddc 11149 − cmin 11482 Basecbs 17205 +gcplusg 17258 Scalarcsca 17261 ·𝑖cip 17263 -gcsg 18922 PreHilcphl 21613 ℂModcclm 25074 ℂPreHilccph 25179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-addf 11225 ax-mulf 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4906 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12516 df-z 12602 df-dec 12721 df-uz 12866 df-fz 13530 df-seq 14013 df-exp 14073 df-struct 17141 df-sets 17158 df-slot 17176 df-ndx 17188 df-base 17206 df-ress 17235 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-0g 17448 df-mgm 18625 df-sgrp 18704 df-mnd 18720 df-mhm 18765 df-grp 18923 df-minusg 18924 df-sbg 18925 df-subg 19110 df-ghm 19200 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20309 df-dvdsr 20332 df-unit 20333 df-rhm 20447 df-subrg 20546 df-drng 20702 df-staf 20811 df-srng 20812 df-lmod 20831 df-lmhm 20993 df-lvec 21074 df-sra 21144 df-rgmod 21145 df-cnfld 21337 df-phl 21615 df-nlm 24580 df-clm 25075 df-cph 25181 |
This theorem is referenced by: nmparlem 25252 cphipval2 25254 |
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