Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ipdir.f | ⊢ 𝐾 = (Base‘𝐹) |
ipass.s | ⊢ · = ( ·𝑠 ‘𝑊) |
ipass.p | ⊢ × = (.r‘𝐹) |
Ref | Expression |
---|---|
ipass | ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlsrng.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | phllmhm.h | . . . . 5 ⊢ , = (·𝑖‘𝑊) | |
3 | phllmhm.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2622 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) | |
5 | 1, 2, 3, 4 | phllmhm 19977 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐶 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
6 | 5 | 3ad2antr3 1228 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
7 | simpr1 1067 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
8 | simpr2 1068 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
9 | ipdir.f | . . . 4 ⊢ 𝐾 = (Base‘𝐹) | |
10 | ipass.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
11 | ipass.p | . . . . 5 ⊢ × = (.r‘𝐹) | |
12 | rlmvsca 19202 | . . . . 5 ⊢ (.r‘𝐹) = ( ·𝑠 ‘(ringLMod‘𝐹)) | |
13 | 11, 12 | eqtri 2644 | . . . 4 ⊢ × = ( ·𝑠 ‘(ringLMod‘𝐹)) |
14 | 1, 9, 3, 10, 13 | lmhmlin 19035 | . . 3 ⊢ (((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘(𝐴 · 𝐵)) = (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵))) |
15 | 6, 7, 8, 14 | syl3anc 1326 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘(𝐴 · 𝐵)) = (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵))) |
16 | phllmod 19975 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
17 | 16 | adantr 481 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ LMod) |
18 | 3, 1, 10, 9 | lmodvscl 18880 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
19 | 17, 7, 8, 18 | syl3anc 1326 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 · 𝐵) ∈ 𝑉) |
20 | oveq1 6657 | . . . 4 ⊢ (𝑥 = (𝐴 · 𝐵) → (𝑥 , 𝐶) = ((𝐴 · 𝐵) , 𝐶)) | |
21 | ovex 6678 | . . . 4 ⊢ (𝑥 , 𝐶) ∈ V | |
22 | 20, 4, 21 | fvmpt3i 6287 | . . 3 ⊢ ((𝐴 · 𝐵) ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘(𝐴 · 𝐵)) = ((𝐴 · 𝐵) , 𝐶)) |
23 | 19, 22 | syl 17 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘(𝐴 · 𝐵)) = ((𝐴 · 𝐵) , 𝐶)) |
24 | oveq1 6657 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 , 𝐶) = (𝐵 , 𝐶)) | |
25 | 24, 4, 21 | fvmpt3i 6287 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵) = (𝐵 , 𝐶)) |
26 | 8, 25 | syl 17 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵) = (𝐵 , 𝐶)) |
27 | 26 | oveq2d 6666 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵)) = (𝐴 × (𝐵 , 𝐶))) |
28 | 15, 23, 27 | 3eqtr3d 2664 | 1 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 ·𝑖cip 15946 LModclmod 18863 LMHom clmhm 19019 ringLModcrglmod 19169 PreHilcphl 19969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-ndx 15860 df-slot 15861 df-sets 15864 df-vsca 15958 df-ip 15959 df-lmod 18865 df-lmhm 19022 df-lvec 19103 df-sra 19172 df-rgmod 19173 df-phl 19971 |
This theorem is referenced by: ipassr 19991 ocvlss 20016 cphass 23011 ipcau2 23033 tchcphlem2 23035 |
Copyright terms: Public domain | W3C validator |