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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 2731 | . . . . 5 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) | |
5 | 1, 2, 3, 4 | phllmhm 21494 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐶 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
6 | 5 | 3ad2antr3 1189 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
7 | simpr1 1193 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝐾) | |
8 | simpr2 1194 | . . 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 21057 | . . . . 5 ⊢ (.r‘𝐹) = ( ·𝑠 ‘(ringLMod‘𝐹)) | |
13 | 11, 12 | eqtri 2759 | . . . 4 ⊢ × = ( ·𝑠 ‘(ringLMod‘𝐹)) |
14 | 1, 9, 3, 10, 13 | lmhmlin 20878 | . . 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 1370 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘(𝐴 · 𝐵)) = (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵))) |
16 | phllmod 21492 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
17 | 16 | adantr 480 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ LMod) |
18 | 3, 1, 10, 9 | lmodvscl 20720 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → (𝐴 · 𝐵) ∈ 𝑉) |
19 | 17, 7, 8, 18 | syl3anc 1370 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 · 𝐵) ∈ 𝑉) |
20 | oveq1 7419 | . . . 4 ⊢ (𝑥 = (𝐴 · 𝐵) → (𝑥 , 𝐶) = ((𝐴 · 𝐵) , 𝐶)) | |
21 | ovex 7445 | . . . 4 ⊢ (𝑥 , 𝐶) ∈ V | |
22 | 20, 4, 21 | fvmpt3i 7003 | . . 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 7419 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑥 , 𝐶) = (𝐵 , 𝐶)) | |
25 | 24, 4, 21 | fvmpt3i 7003 | . . . 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 7428 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 × ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐶))‘𝐵)) = (𝐴 × (𝐵 , 𝐶))) |
28 | 15, 23, 27 | 3eqtr3d 2779 | 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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 .rcmulr 17205 Scalarcsca 17207 ·𝑠 cvsca 17208 ·𝑖cip 17209 LModclmod 20702 LMHom clmhm 20862 ringLModcrglmod 21015 PreHilcphl 21486 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-sets 17104 df-slot 17122 df-ndx 17134 df-sca 17220 df-vsca 17221 df-ip 17222 df-lmod 20704 df-lmhm 20865 df-lvec 20946 df-sra 21018 df-rgmod 21019 df-phl 21488 |
This theorem is referenced by: ipassr 21508 phlssphl 21521 ocvlss 21534 cphass 25058 ipcau2 25081 tcphcphlem2 25083 |
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