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Mirrors > Home > MPE Home > Th. List > dipassr | Structured version Visualization version GIF version |
Description: "Associative" law for second argument of inner product (compare dipass 30366). (Contributed by NM, 22-Nov-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ipass.1 | β’ π = (BaseSetβπ) |
ipass.4 | β’ π = ( Β·π OLD βπ) |
ipass.7 | β’ π = (Β·πOLDβπ) |
Ref | Expression |
---|---|
dipassr | β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (π΄π(π΅ππΆ)) = ((ββπ΅) Β· (π΄ππΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anrot 1099 | . . . 4 β’ ((π΄ β π β§ π΅ β β β§ πΆ β π) β (π΅ β β β§ πΆ β π β§ π΄ β π)) | |
2 | ipass.1 | . . . . 5 β’ π = (BaseSetβπ) | |
3 | ipass.4 | . . . . 5 β’ π = ( Β·π OLD βπ) | |
4 | ipass.7 | . . . . 5 β’ π = (Β·πOLDβπ) | |
5 | 2, 3, 4 | dipass 30366 | . . . 4 β’ ((π β CPreHilOLD β§ (π΅ β β β§ πΆ β π β§ π΄ β π)) β ((π΅ππΆ)ππ΄) = (π΅ Β· (πΆππ΄))) |
6 | 1, 5 | sylan2b 593 | . . 3 β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β ((π΅ππΆ)ππ΄) = (π΅ Β· (πΆππ΄))) |
7 | 6 | fveq2d 6895 | . 2 β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ((π΅ππΆ)ππ΄)) = (ββ(π΅ Β· (πΆππ΄)))) |
8 | phnv 30335 | . . 3 β’ (π β CPreHilOLD β π β NrmCVec) | |
9 | simpl 482 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β π β NrmCVec) | |
10 | 2, 3 | nvscl 30147 | . . . . 5 β’ ((π β NrmCVec β§ π΅ β β β§ πΆ β π) β (π΅ππΆ) β π) |
11 | 10 | 3adant3r1 1181 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (π΅ππΆ) β π) |
12 | simpr1 1193 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β π΄ β π) | |
13 | 2, 4 | dipcj 30235 | . . . 4 β’ ((π β NrmCVec β§ (π΅ππΆ) β π β§ π΄ β π) β (ββ((π΅ππΆ)ππ΄)) = (π΄π(π΅ππΆ))) |
14 | 9, 11, 12, 13 | syl3anc 1370 | . . 3 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ((π΅ππΆ)ππ΄)) = (π΄π(π΅ππΆ))) |
15 | 8, 14 | sylan 579 | . 2 β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ((π΅ππΆ)ππ΄)) = (π΄π(π΅ππΆ))) |
16 | simpr2 1194 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β π΅ β β) | |
17 | 2, 4 | dipcl 30233 | . . . . . . 7 β’ ((π β NrmCVec β§ πΆ β π β§ π΄ β π) β (πΆππ΄) β β) |
18 | 17 | 3com23 1125 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ πΆ β π) β (πΆππ΄) β β) |
19 | 18 | 3adant3r2 1182 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (πΆππ΄) β β) |
20 | 16, 19 | cjmuld 15173 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ(π΅ Β· (πΆππ΄))) = ((ββπ΅) Β· (ββ(πΆππ΄)))) |
21 | 2, 4 | dipcj 30235 | . . . . . . 7 β’ ((π β NrmCVec β§ πΆ β π β§ π΄ β π) β (ββ(πΆππ΄)) = (π΄ππΆ)) |
22 | 21 | 3com23 1125 | . . . . . 6 β’ ((π β NrmCVec β§ π΄ β π β§ πΆ β π) β (ββ(πΆππ΄)) = (π΄ππΆ)) |
23 | 22 | 3adant3r2 1182 | . . . . 5 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ(πΆππ΄)) = (π΄ππΆ)) |
24 | 23 | oveq2d 7428 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β ((ββπ΅) Β· (ββ(πΆππ΄))) = ((ββπ΅) Β· (π΄ππΆ))) |
25 | 20, 24 | eqtrd 2771 | . . 3 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ(π΅ Β· (πΆππ΄))) = ((ββπ΅) Β· (π΄ππΆ))) |
26 | 8, 25 | sylan 579 | . 2 β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (ββ(π΅ Β· (πΆππ΄))) = ((ββπ΅) Β· (π΄ππΆ))) |
27 | 7, 15, 26 | 3eqtr3d 2779 | 1 β’ ((π β CPreHilOLD β§ (π΄ β π β§ π΅ β β β§ πΆ β π)) β (π΄π(π΅ππΆ)) = ((ββπ΅) Β· (π΄ππΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βcfv 6543 (class class class)co 7412 βcc 11111 Β· cmul 11118 βccj 15048 NrmCVeccnv 30105 BaseSetcba 30107 Β·π OLD cns 30108 Β·πOLDcdip 30221 CPreHilOLDccphlo 30333 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7728 ax-inf2 9639 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-cn 22952 df-cnp 22953 df-t1 23039 df-haus 23040 df-tx 23287 df-hmeo 23480 df-xms 24047 df-ms 24048 df-tms 24049 df-grpo 30014 df-gid 30015 df-ginv 30016 df-gdiv 30017 df-ablo 30066 df-vc 30080 df-nv 30113 df-va 30116 df-ba 30117 df-sm 30118 df-0v 30119 df-vs 30120 df-nmcv 30121 df-ims 30122 df-dip 30222 df-ph 30334 |
This theorem is referenced by: dipassr2 30368 |
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