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Mirrors > Home > MPE Home > Th. List > ipass | Structured version Visualization version GIF version |
Description: Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | β’ πΉ = (Scalarβπ) |
phllmhm.h | β’ , = (Β·πβπ) |
phllmhm.v | β’ π = (Baseβπ) |
ipdir.f | β’ πΎ = (BaseβπΉ) |
ipass.s | β’ Β· = ( Β·π βπ) |
ipass.p | β’ Γ = (.rβπΉ) |
Ref | Expression |
---|---|
ipass | β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((π΄ Β· π΅) , πΆ) = (π΄ Γ (π΅ , πΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlsrng.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
2 | phllmhm.h | . . . . 5 β’ , = (Β·πβπ) | |
3 | phllmhm.v | . . . . 5 β’ π = (Baseβπ) | |
4 | eqid 2732 | . . . . 5 β’ (π₯ β π β¦ (π₯ , πΆ)) = (π₯ β π β¦ (π₯ , πΆ)) | |
5 | 1, 2, 3, 4 | phllmhm 21184 | . . . 4 β’ ((π β PreHil β§ πΆ β π) β (π₯ β π β¦ (π₯ , πΆ)) β (π LMHom (ringLModβπΉ))) |
6 | 5 | 3ad2antr3 1190 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β (π₯ β π β¦ (π₯ , πΆ)) β (π LMHom (ringLModβπΉ))) |
7 | simpr1 1194 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β π΄ β πΎ) | |
8 | simpr2 1195 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β π΅ β π) | |
9 | ipdir.f | . . . 4 β’ πΎ = (BaseβπΉ) | |
10 | ipass.s | . . . 4 β’ Β· = ( Β·π βπ) | |
11 | ipass.p | . . . . 5 β’ Γ = (.rβπΉ) | |
12 | rlmvsca 20823 | . . . . 5 β’ (.rβπΉ) = ( Β·π β(ringLModβπΉ)) | |
13 | 11, 12 | eqtri 2760 | . . . 4 β’ Γ = ( Β·π β(ringLModβπΉ)) |
14 | 1, 9, 3, 10, 13 | lmhmlin 20645 | . . 3 β’ (((π₯ β π β¦ (π₯ , πΆ)) β (π LMHom (ringLModβπΉ)) β§ π΄ β πΎ β§ π΅ β π) β ((π₯ β π β¦ (π₯ , πΆ))β(π΄ Β· π΅)) = (π΄ Γ ((π₯ β π β¦ (π₯ , πΆ))βπ΅))) |
15 | 6, 7, 8, 14 | syl3anc 1371 | . 2 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((π₯ β π β¦ (π₯ , πΆ))β(π΄ Β· π΅)) = (π΄ Γ ((π₯ β π β¦ (π₯ , πΆ))βπ΅))) |
16 | phllmod 21182 | . . . . 5 β’ (π β PreHil β π β LMod) | |
17 | 16 | adantr 481 | . . . 4 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β π β LMod) |
18 | 3, 1, 10, 9 | lmodvscl 20488 | . . . 4 β’ ((π β LMod β§ π΄ β πΎ β§ π΅ β π) β (π΄ Β· π΅) β π) |
19 | 17, 7, 8, 18 | syl3anc 1371 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β (π΄ Β· π΅) β π) |
20 | oveq1 7415 | . . . 4 β’ (π₯ = (π΄ Β· π΅) β (π₯ , πΆ) = ((π΄ Β· π΅) , πΆ)) | |
21 | ovex 7441 | . . . 4 β’ (π₯ , πΆ) β V | |
22 | 20, 4, 21 | fvmpt3i 7003 | . . 3 β’ ((π΄ Β· π΅) β π β ((π₯ β π β¦ (π₯ , πΆ))β(π΄ Β· π΅)) = ((π΄ Β· π΅) , πΆ)) |
23 | 19, 22 | syl 17 | . 2 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((π₯ β π β¦ (π₯ , πΆ))β(π΄ Β· π΅)) = ((π΄ Β· π΅) , πΆ)) |
24 | oveq1 7415 | . . . . 5 β’ (π₯ = π΅ β (π₯ , πΆ) = (π΅ , πΆ)) | |
25 | 24, 4, 21 | fvmpt3i 7003 | . . . 4 β’ (π΅ β π β ((π₯ β π β¦ (π₯ , πΆ))βπ΅) = (π΅ , πΆ)) |
26 | 8, 25 | syl 17 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((π₯ β π β¦ (π₯ , πΆ))βπ΅) = (π΅ , πΆ)) |
27 | 26 | oveq2d 7424 | . 2 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β (π΄ Γ ((π₯ β π β¦ (π₯ , πΆ))βπ΅)) = (π΄ Γ (π΅ , πΆ))) |
28 | 15, 23, 27 | 3eqtr3d 2780 | 1 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β ((π΄ Β· π΅) , πΆ) = (π΄ Γ (π΅ , πΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 β¦ cmpt 5231 βcfv 6543 (class class class)co 7408 Basecbs 17143 .rcmulr 17197 Scalarcsca 17199 Β·π cvsca 17200 Β·πcip 17201 LModclmod 20470 LMHom clmhm 20629 ringLModcrglmod 20781 PreHilcphl 21176 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 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-op 4635 df-uni 4909 df-iun 4999 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-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-riota 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7855 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-sets 17096 df-slot 17114 df-ndx 17126 df-sca 17212 df-vsca 17213 df-ip 17214 df-lmod 20472 df-lmhm 20632 df-lvec 20713 df-sra 20784 df-rgmod 20785 df-phl 21178 |
This theorem is referenced by: ipassr 21198 phlssphl 21211 ocvlss 21224 cphass 24727 ipcau2 24750 tcphcphlem2 24752 |
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