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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 22 | . . 3 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) | |
| 2 | 1 | 3com13 1124 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) |
| 3 | id 22 | . . . . . 6 ⊢ ((𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) | |
| 4 | 3 | 3com12 1123 | . . . . 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 30777 | . . . . 5 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐵𝑀𝐶)𝑃𝐴) = ((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) |
| 9 | 4, 8 | sylan2 593 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐵𝑀𝐶)𝑃𝐴) = ((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) |
| 10 | 9 | fveq2d 6862 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴)))) |
| 11 | phnv 30743 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
| 12 | simpl 482 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝑈 ∈ NrmCVec) | |
| 13 | 5, 6 | nvmcl 30575 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 14 | 13 | 3com23 1126 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 15 | 14 | 3adant3r3 1185 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 16 | simpr3 1197 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
| 17 | 5, 7 | dipcj 30643 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐵𝑀𝐶) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 18 | 12, 15, 16, 17 | syl3anc 1373 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 19 | 11, 18 | sylan 580 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 20 | 5, 7 | dipcl 30641 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝑃𝐴) ∈ ℂ) |
| 21 | 20 | 3adant3r1 1183 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵𝑃𝐴) ∈ ℂ) |
| 22 | 5, 7 | dipcl 30641 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶𝑃𝐴) ∈ ℂ) |
| 23 | 22 | 3adant3r2 1184 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐶𝑃𝐴) ∈ ℂ) |
| 24 | cjsub 15115 | . . . . . 6 ⊢ (((𝐵𝑃𝐴) ∈ ℂ ∧ (𝐶𝑃𝐴) ∈ ℂ) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴)))) | |
| 25 | 21, 23, 24 | syl2anc 584 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴)))) |
| 26 | 5, 7 | dipcj 30643 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)) |
| 27 | 26 | 3adant3r1 1183 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)) |
| 28 | 5, 7 | dipcj 30643 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐶𝑃𝐴)) = (𝐴𝑃𝐶)) |
| 29 | 28 | 3adant3r2 1184 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘(𝐶𝑃𝐴)) = (𝐴𝑃𝐶)) |
| 30 | 27, 29 | oveq12d 7405 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 31 | 25, 30 | eqtrd 2764 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 32 | 11, 31 | sylan 580 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 33 | 10, 19, 32 | 3eqtr3d 2772 | . 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 34 | 2, 33 | sylan2 593 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 − cmin 11405 ∗ccj 15062 NrmCVeccnv 30513 BaseSetcba 30515 −𝑣 cnsb 30518 ·𝑖OLDcdip 30629 CPreHilOLDccphlo 30741 |
| 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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146 ax-addf 11147 ax-mulf 11148 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-supp 8140 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-ixp 8871 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-fsupp 9313 df-fi 9362 df-sup 9393 df-inf 9394 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-ioo 13310 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-rest 17385 df-topn 17386 df-0g 17404 df-gsum 17405 df-topgen 17406 df-pt 17407 df-prds 17410 df-xrs 17465 df-qtop 17470 df-imas 17471 df-xps 17473 df-mre 17547 df-mrc 17548 df-acs 17550 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19249 df-cmn 19712 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-cn 23114 df-cnp 23115 df-t1 23201 df-haus 23202 df-tx 23449 df-hmeo 23642 df-xms 24208 df-ms 24209 df-tms 24210 df-grpo 30422 df-gid 30423 df-ginv 30424 df-gdiv 30425 df-ablo 30474 df-vc 30488 df-nv 30521 df-va 30524 df-ba 30525 df-sm 30526 df-0v 30527 df-vs 30528 df-nmcv 30529 df-ims 30530 df-dip 30630 df-ph 30742 |
| This theorem is referenced by: siilem1 30780 ip2eqi 30785 |
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