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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 2727 | . . . . 5 β’ (π₯ β π β¦ (π₯ , πΆ)) = (π₯ β π β¦ (π₯ , πΆ)) | |
5 | 1, 2, 3, 4 | phllmhm 21557 | . . . 4 β’ ((π β PreHil β§ πΆ β π) β (π₯ β π β¦ (π₯ , πΆ)) β (π LMHom (ringLModβπΉ))) |
6 | 5 | 3ad2antr3 1188 | . . 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 1192 | . . 3 β’ ((π β PreHil β§ (π΄ β πΎ β§ π΅ β π β§ πΆ β π)) β π΄ β πΎ) | |
8 | simpr2 1193 | . . 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 21086 | . . . . 5 β’ (.rβπΉ) = ( Β·π β(ringLModβπΉ)) | |
13 | 11, 12 | eqtri 2755 | . . . 4 β’ Γ = ( Β·π β(ringLModβπΉ)) |
14 | 1, 9, 3, 10, 13 | lmhmlin 20913 | . . 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 1369 | . 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 21555 | . . . . 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 20754 | . . . 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 1369 | . . 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 7421 | . . . 4 β’ (π₯ = (π΄ Β· π΅) β (π₯ , πΆ) = ((π΄ Β· π΅) , πΆ)) | |
21 | ovex 7447 | . . . 4 β’ (π₯ , πΆ) β V | |
22 | 20, 4, 21 | fvmpt3i 7004 | . . 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 7421 | . . . . 5 β’ (π₯ = π΅ β (π₯ , πΆ) = (π΅ , πΆ)) | |
25 | 24, 4, 21 | fvmpt3i 7004 | . . . 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 7430 | . 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 2775 | 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 1085 = wceq 1534 β wcel 2099 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 Basecbs 17173 .rcmulr 17227 Scalarcsca 17229 Β·π cvsca 17230 Β·πcip 17231 LModclmod 20736 LMHom clmhm 20897 ringLModcrglmod 21050 PreHilcphl 21549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-sets 17126 df-slot 17144 df-ndx 17156 df-sca 17242 df-vsca 17243 df-ip 17244 df-lmod 20738 df-lmhm 20900 df-lvec 20981 df-sra 21051 df-rgmod 21052 df-phl 21551 |
This theorem is referenced by: ipassr 21571 phlssphl 21584 ocvlss 21597 cphass 25132 ipcau2 25155 tcphcphlem2 25157 |
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