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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 31230 | . . . . 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 6892 | . . 3 ⊢ ((𝑈 ∈ CPreHilOLD ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (∗‘((𝐵𝑀𝐶)𝑃𝐴)) = (∗‘((𝐵𝑃𝐴) − (𝐶𝑃𝐴)))) |
| 11 | phnv 31196 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
| 12 | simpl 488 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → 𝑈 ∈ NrmCVec) | |
| 13 | 5, 6 | nvmcl 31028 | . . . . . . 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 31096 | . . . . 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 31094 | . . . . . . 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 31094 | . . . . . . 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 15226 | . . . . . 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 31096 | . . . . . . 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 31096 | . . . . . . 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 7441 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((∗‘(𝐵𝑃𝐴)) − (∗‘(𝐶𝑃𝐴))) = ((𝐴𝑃𝐵) − (𝐴𝑃𝐶))) |
| 31 | 25, 30 | eqtrd 2801 | . . . 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 2809 | . 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 6543 (class class class)co 7423 ℂcc 11116 − cmin 11459 ∗ccj 15173 NrmCVeccnv 30966 BaseSetcba 30968 −𝑣 cnsb 30971 ·𝑖OLDcdip 31082 CPreHilOLDccphlo 31194 |
| 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 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-inf2 9620 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197 ax-mulf 11198 |
| 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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-card 9944 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-icc 13397 df-fz 13554 df-fzo 13702 df-seq 14058 df-exp 14118 df-hash 14387 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-sum 15764 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-xrs 17581 df-qtop 17586 df-imas 17587 df-xps 17589 df-mre 17663 df-mrc 17664 df-acs 17666 df-mgm 18723 df-sgrp 18802 df-mnd 18818 df-submnd 18867 df-mulg 19159 df-cntz 19412 df-cmn 19877 df-psmet 21544 df-xmet 21545 df-met 21546 df-bl 21547 df-mopn 21548 df-cnfld 21553 df-top 23081 df-topon 23098 df-topsp 23120 df-bases 23133 df-cld 23206 df-ntr 23207 df-cls 23208 df-cn 23414 df-cnp 23415 df-t1 23501 df-haus 23502 df-tx 23749 df-hmeo 23942 df-xms 24507 df-ms 24508 df-tms 24509 df-grpo 30875 df-gid 30876 df-ginv 30877 df-gdiv 30878 df-ablo 30927 df-vc 30941 df-nv 30974 df-va 30977 df-ba 30978 df-sm 30979 df-0v 30980 df-vs 30981 df-nmcv 30982 df-ims 30983 df-dip 31083 df-ph 31195 |
| This theorem is used by: siilem1 31233 ip2eqi 31238 |
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