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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 2724 | . . . . 5 β’ (π₯ β π β¦ (π₯ , πΆ)) = (π₯ β π β¦ (π₯ , πΆ)) | |
5 | 1, 2, 3, 4 | phllmhm 21493 | . . . 4 β’ ((π β PreHil β§ πΆ β π) β (π₯ β π β¦ (π₯ , πΆ)) β (π LMHom (ringLModβπΉ))) |
6 | 5 | 3ad2antr3 1187 | . . 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 1191 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β π΄ β πΎ) | |
8 | simpr2 1192 | . . 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 21046 | . . . . 5 β’ (.rβπΉ) = ( Β·π β(ringLModβπΉ)) | |
13 | 11, 12 | eqtri 2752 | . . . 4 β’ Γ = ( Β·π β(ringLModβπΉ)) |
14 | 1, 9, 3, 10, 13 | lmhmlin 20873 | . . 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 1368 | . 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 21491 | . . . . 5 β’ (π β PreHil β π β LMod) | |
17 | 16 | adantr 480 | . . . 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 20714 | . . . 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 1368 | . . 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 7408 | . . . 4 β’ (π₯ = (π΄ Β· π΅) β (π₯ , πΆ) = ((π΄ Β· π΅) , πΆ)) | |
21 | ovex 7434 | . . . 4 β’ (π₯ , πΆ) β V | |
22 | 20, 4, 21 | fvmpt3i 6993 | . . 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 7408 | . . . . 5 β’ (π₯ = π΅ β (π₯ , πΆ) = (π΅ , πΆ)) | |
25 | 24, 4, 21 | fvmpt3i 6993 | . . . 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 7417 | . 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 2772 | 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 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β¦ cmpt 5221 βcfv 6533 (class class class)co 7401 Basecbs 17143 .rcmulr 17197 Scalarcsca 17199 Β·π cvsca 17200 Β·πcip 17201 LModclmod 20696 LMHom clmhm 20857 ringLModcrglmod 21010 PreHilcphl 21485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 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 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-sets 17096 df-slot 17114 df-ndx 17126 df-sca 17212 df-vsca 17213 df-ip 17214 df-lmod 20698 df-lmhm 20860 df-lvec 20941 df-sra 21011 df-rgmod 21012 df-phl 21487 |
This theorem is referenced by: ipassr 21507 phlssphl 21520 ocvlss 21533 cphass 25061 ipcau2 25084 tcphcphlem2 25086 |
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