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| Mirrors > Home > MPE Home > Th. List > dipsubdi | Structured version Visualization version GIF version | ||
| Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| ipsubdir.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
| ipsubdir.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
| ipsubdir.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
| Ref | Expression |
|---|---|
| dipsubdi | ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 23 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) | |
| 2 | 1 | 3com13 1142 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
| 3 | id 23 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) | |
| 4 | 3 | 3com12 1141 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
| 5 | ipsubdir.1 | . . . . . 6 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 6 | ipsubdir.3 | . . . . . 6 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
| 7 | ipsubdir.7 | . . . . . 6 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
| 8 | 5, 6, 7 | dipsubdir 31229 | . . . . 5 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐵𝑀𝐶)𝑃𝐴) = ((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) |
| 9 | 4, 8 | sylan2 605 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐵𝑀𝐶)𝑃𝐴) = ((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) |
| 10 | 9 | fveq2d 6889 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴)))) |
| 11 | phnv 31195 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
| 12 | simpl 488 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝑈 ∈ NrmCVec) | |
| 13 | 5, 6 | nvmcl 31027 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 14 | 13 | 3com23 1144 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 15 | 14 | 3adant3r3 1203 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 16 | simpr3 1215 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
| 17 | 5, 7 | dipcj 31095 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐵𝑀𝐶) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 18 | 12, 15, 16, 17 | syl3anc 1398 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 19 | 11, 18 | sylan 592 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 20 | 5, 7 | dipcl 31093 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝑃𝐴) ∈ ℂ) |
| 21 | 20 | 3adant3r1 1201 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵𝑃𝐴) ∈ ℂ) |
| 22 | 5, 7 | dipcl 31093 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶𝑃𝐴) ∈ ℂ) |
| 23 | 22 | 3adant3r2 1202 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐶𝑃𝐴) ∈ ℂ) |
| 24 | cjsub 15219 | . . . . . 6 ⊢ (((𝐵𝑃𝐴) ∈ ℂ ∧ (𝐶𝑃𝐴) ∈ ℂ) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴)))) | |
| 25 | 21, 23, 24 | syl2anc 596 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴)))) |
| 26 | 5, 7 | dipcj 31095 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)) |
| 27 | 26 | 3adant3r1 1201 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)) |
| 28 | 5, 7 | dipcj 31095 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐶𝑃𝐴)) = (𝐴𝑃𝐶)) |
| 29 | 28 | 3adant3r2 1202 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘(𝐶𝑃𝐴)) = (𝐴𝑃𝐶)) |
| 30 | 27, 29 | oveq12d 7434 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 31 | 25, 30 | eqtrd 2800 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 32 | 11, 31 | sylan 592 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 33 | 10, 19, 32 | 3eqtr3d 2808 | . 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 34 | 2, 33 | sylan2 605 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ‘cfv 6540 (class class class)co 7416 ℂcc 11109 − cmin 11452 ∗ccj 15166 NrmCVeccnv 30965 BaseSetcba 30967 −𝑣 cnsb 30970 ·𝑖OLDcdip 31081 CPreHilOLDccphlo 31193 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190 ax-mulf 11191 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-q 12984 df-rp 13028 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13387 df-icc 13390 df-fz 13547 df-fzo 13695 df-seq 14051 df-exp 14111 df-hash 14380 df-cj 15169 df-re 15170 df-im 15171 df-sqrt 15305 df-abs 15306 df-clim 15558 df-sum 15757 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-starv 17342 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-unif 17350 df-hom 17351 df-cco 17352 df-rest 17492 df-topn 17493 df-0g 17511 df-gsum 17512 df-topgen 17513 df-pt 17514 df-prds 17517 df-xrs 17573 df-qtop 17578 df-imas 17579 df-xps 17581 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-submnd 18865 df-mulg 19157 df-cntz 19410 df-cmn 19875 df-psmet 21543 df-xmet 21544 df-met 21545 df-bl 21546 df-mopn 21547 df-cnfld 21552 df-top 23080 df-topon 23097 df-topsp 23119 df-bases 23132 df-cld 23205 df-ntr 23206 df-cls 23207 df-cn 23413 df-cnp 23414 df-t1 23500 df-haus 23501 df-tx 23748 df-hmeo 23941 df-xms 24506 df-ms 24507 df-tms 24508 df-grpo 30874 df-gid 30875 df-ginv 30876 df-gdiv 30877 df-ablo 30926 df-vc 30940 df-nv 30973 df-va 30976 df-ba 30977 df-sm 30978 df-0v 30979 df-vs 30980 df-nmcv 30981 df-ims 30982 df-dip 31082 df-ph 31194 |
| This theorem is used by: siilem1 31232 ip2eqi 31237 |
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