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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 | ⊢ 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 2821 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) | |
5 | 1, 2, 3, 4 | phllmhm 20776 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐶 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
6 | 5 | 3ad2antr3 1186 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
7 | simpr1 1190 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
8 | simpr2 1191 | . . 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 19974 | . . . . 5 ⊢ (.r‘𝐹) = ( ·𝑠 ‘(ringLMod‘𝐹)) | |
13 | 11, 12 | eqtri 2844 | . . . 4 ⊢ × = ( ·𝑠 ‘(ringLMod‘𝐹)) |
14 | 1, 9, 3, 10, 13 | lmhmlin 19807 | . . 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 1367 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘(𝐴 · 𝐵)) = (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵))) |
16 | phllmod 20774 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
17 | 16 | adantr 483 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ LMod) |
18 | 3, 1, 10, 9 | lmodvscl 19651 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
19 | 17, 7, 8, 18 | syl3anc 1367 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 · 𝐵) ∈ 𝑉) |
20 | oveq1 7163 | . . . 4 ⊢ (𝑥 = (𝐴 · 𝐵) → (𝑥 , 𝐶) = ((𝐴 · 𝐵) , 𝐶)) | |
21 | ovex 7189 | . . . 4 ⊢ (𝑥 , 𝐶) ∈ V | |
22 | 20, 4, 21 | fvmpt3i 6773 | . . 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 7163 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 , 𝐶) = (𝐵 , 𝐶)) | |
25 | 24, 4, 21 | fvmpt3i 6773 | . . . 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 7172 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵)) = (𝐴 × (𝐵 , 𝐶))) |
28 | 15, 23, 27 | 3eqtr3d 2864 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 ·𝑖cip 16570 LModclmod 19634 LMHom clmhm 19791 ringLModcrglmod 19941 PreHilcphl 20768 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-ndx 16486 df-slot 16487 df-sets 16490 df-vsca 16582 df-ip 16583 df-lmod 19636 df-lmhm 19794 df-lvec 19875 df-sra 19944 df-rgmod 19945 df-phl 20770 |
This theorem is referenced by: ipassr 20790 phlssphl 20803 ocvlss 20816 cphass 23815 ipcau2 23837 tcphcphlem2 23839 |
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