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Mirrors > Home > MPE Home > Th. List > ip2di | Structured version Visualization version GIF version |
Description: Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ipdir.g | ⊢ + = (+g‘𝑊) |
ipdir.p | ⊢ ⨣ = (+g‘𝐹) |
ip2di.1 | ⊢ (𝜑 → 𝑊 ∈ PreHil) |
ip2di.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
ip2di.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑉) |
ip2di.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
ip2di.5 | ⊢ (𝜑 → 𝐷 ∈ 𝑉) |
Ref | Expression |
---|---|
ip2di | ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ip2di.1 | . . 3 ⊢ (𝜑 → 𝑊 ∈ PreHil) | |
2 | ip2di.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
3 | ip2di.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉) | |
4 | phllmod 20373 | . . . . 5 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
5 | 1, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ LMod) |
6 | ip2di.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
7 | ip2di.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
8 | phllmhm.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
9 | ipdir.g | . . . . 5 ⊢ + = (+g‘𝑊) | |
10 | 8, 9 | lmodvacl 19269 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉) |
11 | 5, 6, 7, 10 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ 𝑉) |
12 | phlsrng.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
13 | phllmhm.h | . . . 4 ⊢ , = (·𝑖‘𝑊) | |
14 | ipdir.p | . . . 4 ⊢ ⨣ = (+g‘𝐹) | |
15 | 12, 13, 8, 9, 14 | ipdir 20382 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ (𝐶 + 𝐷) ∈ 𝑉)) → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷)))) |
16 | 1, 2, 3, 11, 15 | syl13anc 1440 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷)))) |
17 | 12, 13, 8, 9, 14 | ipdi 20383 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
18 | 1, 2, 6, 7, 17 | syl13anc 1440 | . . 3 ⊢ (𝜑 → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
19 | 12, 13, 8, 9, 14 | ipdi 20383 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
20 | 1, 3, 6, 7, 19 | syl13anc 1440 | . . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
21 | 12 | phlsrng 20374 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
22 | srngring 19244 | . . . . . 6 ⊢ (𝐹 ∈ *-Ring → 𝐹 ∈ Ring) | |
23 | ringcmn 18968 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ CMnd) | |
24 | 1, 21, 22, 23 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ CMnd) |
25 | eqid 2778 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 12, 13, 8, 25 | ipcl 20376 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
27 | 1, 3, 6, 26 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
28 | 12, 13, 8, 25 | ipcl 20376 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
29 | 1, 3, 7, 28 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
30 | 25, 14 | cmncom 18595 | . . . . 5 ⊢ ((𝐹 ∈ CMnd ∧ (𝐵 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐵 , 𝐷) ∈ (Base‘𝐹)) → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
31 | 24, 27, 29, 30 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
32 | 20, 31 | eqtrd 2814 | . . 3 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
33 | 18, 32 | oveq12d 6940 | . 2 ⊢ (𝜑 → ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))) = (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
34 | 12, 13, 8, 25 | ipcl 20376 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
35 | 1, 2, 6, 34 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
36 | 12, 13, 8, 25 | ipcl 20376 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
37 | 1, 2, 7, 36 | syl3anc 1439 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
38 | 25, 14 | cmn4 18598 | . . 3 ⊢ ((𝐹 ∈ CMnd ∧ ((𝐴 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐴 , 𝐷) ∈ (Base‘𝐹)) ∧ ((𝐵 , 𝐷) ∈ (Base‘𝐹) ∧ (𝐵 , 𝐶) ∈ (Base‘𝐹))) → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
39 | 24, 35, 37, 29, 27, 38 | syl122anc 1447 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
40 | 16, 33, 39 | 3eqtrd 2818 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 +gcplusg 16338 Scalarcsca 16341 ·𝑖cip 16343 CMndccmn 18579 Ringcrg 18934 *-Ringcsr 19236 LModclmod 19255 PreHilcphl 20367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-ip 16356 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-mhm 17721 df-grp 17812 df-minusg 17813 df-ghm 18042 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-rnghom 19104 df-staf 19237 df-srng 19238 df-lmod 19257 df-lmhm 19417 df-lvec 19498 df-sra 19569 df-rgmod 19570 df-phl 20369 |
This theorem is referenced by: cph2di 23414 |
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