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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 20344 | . . . . 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 19240 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉) |
11 | 5, 6, 7, 10 | syl3anc 1494 | . . 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 20353 | . . 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 1495 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷)))) |
17 | 12, 13, 8, 9, 14 | ipdi 20354 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
18 | 1, 2, 6, 7, 17 | syl13anc 1495 | . . 3 ⊢ (𝜑 → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
19 | 12, 13, 8, 9, 14 | ipdi 20354 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
20 | 1, 3, 6, 7, 19 | syl13anc 1495 | . . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
21 | 12 | phlsrng 20345 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
22 | srngring 19215 | . . . . . 6 ⊢ (𝐹 ∈ *-Ring → 𝐹 ∈ Ring) | |
23 | ringcmn 18942 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ CMnd) | |
24 | 1, 21, 22, 23 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ CMnd) |
25 | eqid 2825 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 12, 13, 8, 25 | ipcl 20347 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
27 | 1, 3, 6, 26 | syl3anc 1494 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
28 | 12, 13, 8, 25 | ipcl 20347 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
29 | 1, 3, 7, 28 | syl3anc 1494 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
30 | 25, 14 | cmncom 18569 | . . . . 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 1494 | . . . 4 ⊢ (𝜑 → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
32 | 20, 31 | eqtrd 2861 | . . 3 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
33 | 18, 32 | oveq12d 6928 | . 2 ⊢ (𝜑 → ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))) = (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
34 | 12, 13, 8, 25 | ipcl 20347 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
35 | 1, 2, 6, 34 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
36 | 12, 13, 8, 25 | ipcl 20347 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
37 | 1, 2, 7, 36 | syl3anc 1494 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
38 | 25, 14 | cmn4 18572 | . . 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 1502 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
40 | 16, 33, 39 | 3eqtrd 2865 | 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 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 Scalarcsca 16315 ·𝑖cip 16317 CMndccmn 18553 Ringcrg 18908 *-Ringcsr 19207 LModclmod 19226 PreHilcphl 20338 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-tpos 7622 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-ip 16330 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-ghm 18016 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-rnghom 19078 df-staf 19208 df-srng 19209 df-lmod 19228 df-lmhm 19388 df-lvec 19469 df-sra 19540 df-rgmod 19541 df-phl 20340 |
This theorem is referenced by: cph2di 23383 |
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