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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 1120 | . 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 1119 | . . . . 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 28627 | . . . . 5 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐵𝑀𝐶)𝑃𝐴) = ((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) |
| 9 | 4, 8 | sylan2 594 | . . . 4 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐵𝑀𝐶)𝑃𝐴) = ((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) |
| 10 | 9 | fveq2d 6676 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴)))) |
| 11 | phnv 28593 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
| 12 | simpl 485 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝑈 ∈ NrmCVec) | |
| 13 | 5, 6 | nvmcl 28425 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 14 | 13 | 3com23 1122 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 15 | 14 | 3adant3r3 1180 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵𝑀𝐶) ∈ 𝑋) |
| 16 | simpr3 1192 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝐴 ∈ 𝑋) | |
| 17 | 5, 7 | dipcj 28493 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐵𝑀𝐶) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 18 | 12, 15, 16, 17 | syl3anc 1367 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 19 | 11, 18 | sylan 582 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (𝐴𝑃(𝐵𝑀𝐶))) |
| 20 | 5, 7 | dipcl 28491 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝑃𝐴) ∈ ℂ) |
| 21 | 20 | 3adant3r1 1178 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐵𝑃𝐴) ∈ ℂ) |
| 22 | 5, 7 | dipcl 28491 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐶𝑃𝐴) ∈ ℂ) |
| 23 | 22 | 3adant3r2 1179 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐶𝑃𝐴) ∈ ℂ) |
| 24 | cjsub 14510 | . . . . . 6 ⊢ (((𝐵𝑃𝐴) ∈ ℂ ∧ (𝐶𝑃𝐴) ∈ ℂ) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴)))) | |
| 25 | 21, 23, 24 | syl2anc 586 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴)))) |
| 26 | 5, 7 | dipcj 28493 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)) |
| 27 | 26 | 3adant3r1 1178 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘(𝐵𝑃𝐴)) = (𝐴𝑃𝐵)) |
| 28 | 5, 7 | dipcj 28493 | . . . . . . 7 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (∗‘(𝐶𝑃𝐴)) = (𝐴𝑃𝐶)) |
| 29 | 28 | 3adant3r2 1179 | . . . . . 6 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘(𝐶𝑃𝐴)) = (𝐴𝑃𝐶)) |
| 30 | 27, 29 | oveq12d 7176 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 31 | 25, 30 | eqtrd 2858 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 32 | 11, 31 | sylan 582 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 33 | 10, 19, 32 | 3eqtr3d 2866 | . 2 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 34 | 2, 33 | sylan2 594 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑃(𝐵𝑀𝐶)) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 − cmin 10872 ∗ccj 14457 NrmCVeccnv 28363 BaseSetcba 28365 −𝑣 cnsb 28368 ·𝑖OLDcdip 28479 CPreHilOLDccphlo 28591 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-cn 21837 df-cnp 21838 df-t1 21924 df-haus 21925 df-tx 22172 df-hmeo 22365 df-xms 22932 df-ms 22933 df-tms 22934 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-dip 28480 df-ph 28592 |
| This theorem is referenced by: siilem1 28630 ip2eqi 28635 |
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