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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 21662. (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 25223 | . . . . . 6 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod) | |
| 3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ ℂMod) |
| 4 | eqid 2737 | . . . . . 6 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
| 5 | 4 | clmadd 25107 | . . . . 5 ⊢ (𝑊 ∈ ℂMod → + = (+g‘(Scalar‘𝑊))) |
| 6 | 3, 5 | syl 17 | . . . 4 ⊢ (𝜑 → + = (+g‘(Scalar‘𝑊))) |
| 7 | 6 | oveqd 7448 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐶) + (𝐵 , 𝐷)) = ((𝐴 , 𝐶)(+g‘(Scalar‘𝑊))(𝐵 , 𝐷))) |
| 8 | 6 | oveqd 7448 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) = ((𝐴 , 𝐷)(+g‘(Scalar‘𝑊))(𝐵 , 𝐶))) |
| 9 | 7, 8 | oveq12d 7449 | . 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 25205 | . . . . . 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 2737 | . . . . . 6 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
| 17 | 4, 14, 15, 16 | ipcl 21651 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
| 18 | 11, 12, 13, 17 | syl3anc 1373 | . . . 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 21651 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
| 22 | 11, 19, 20, 21 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
| 23 | 4, 16 | clmacl 25117 | . . . 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 1373 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐶) + (𝐵 , 𝐷)) ∈ (Base‘(Scalar‘𝑊))) |
| 25 | 4, 14, 15, 16 | ipcl 21651 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
| 26 | 11, 12, 20, 25 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘(Scalar‘𝑊))) |
| 27 | 4, 14, 15, 16 | ipcl 21651 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
| 28 | 11, 19, 13, 27 | syl3anc 1373 | . . . 4 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘(Scalar‘𝑊))) |
| 29 | 4, 16 | clmacl 25117 | . . . 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 1373 | . . 3 ⊢ (𝜑 → ((𝐴 , 𝐷) + (𝐵 , 𝐶)) ∈ (Base‘(Scalar‘𝑊))) |
| 31 | 4, 16 | clmsub 25113 | . . 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 1373 | . 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 2737 | . . 3 ⊢ (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑊)) | |
| 35 | eqid 2737 | . . 3 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
| 36 | 4, 14, 15, 33, 34, 35, 11, 12, 19, 13, 20 | ip2subdi 21662 | . 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 2788 | 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 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 + caddc 11158 − cmin 11492 Basecbs 17247 +gcplusg 17297 Scalarcsca 17300 ·𝑖cip 17302 -gcsg 18953 PreHilcphl 21642 ℂModcclm 25095 ℂPreHilccph 25200 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-addf 11234 ax-mulf 11235 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-fz 13548 df-seq 14043 df-exp 14103 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mhm 18796 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-ghm 19231 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-cring 20233 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-rhm 20472 df-subrg 20570 df-drng 20731 df-staf 20840 df-srng 20841 df-lmod 20860 df-lmhm 21021 df-lvec 21102 df-sra 21172 df-rgmod 21173 df-cnfld 21365 df-phl 21644 df-nlm 24599 df-clm 25096 df-cph 25202 |
| This theorem is referenced by: nmparlem 25273 cphipval2 25275 |
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