Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 20702 | . . . . 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 19577 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉) |
11 | 5, 6, 7, 10 | syl3anc 1363 | . . 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 20711 | . . 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 1364 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷)))) |
17 | 12, 13, 8, 9, 14 | ipdi 20712 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
18 | 1, 2, 6, 7, 17 | syl13anc 1364 | . . 3 ⊢ (𝜑 → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
19 | 12, 13, 8, 9, 14 | ipdi 20712 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
20 | 1, 3, 6, 7, 19 | syl13anc 1364 | . . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
21 | 12 | phlsrng 20703 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
22 | srngring 19552 | . . . . . 6 ⊢ (𝐹 ∈ *-Ring → 𝐹 ∈ Ring) | |
23 | ringcmn 19260 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ CMnd) | |
24 | 1, 21, 22, 23 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ CMnd) |
25 | eqid 2818 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 12, 13, 8, 25 | ipcl 20705 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
27 | 1, 3, 6, 26 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
28 | 12, 13, 8, 25 | ipcl 20705 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
29 | 1, 3, 7, 28 | syl3anc 1363 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
30 | 25, 14 | cmncom 18852 | . . . . 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 1363 | . . . 4 ⊢ (𝜑 → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
32 | 20, 31 | eqtrd 2853 | . . 3 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
33 | 18, 32 | oveq12d 7163 | . 2 ⊢ (𝜑 → ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))) = (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
34 | 12, 13, 8, 25 | ipcl 20705 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
35 | 1, 2, 6, 34 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
36 | 12, 13, 8, 25 | ipcl 20705 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
37 | 1, 2, 7, 36 | syl3anc 1363 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
38 | 25, 14 | cmn4 18855 | . . 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 1371 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
40 | 16, 33, 39 | 3eqtrd 2857 | 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 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Scalarcsca 16556 ·𝑖cip 16558 CMndccmn 18835 Ringcrg 19226 *-Ringcsr 19544 LModclmod 19563 PreHilcphl 20696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-grp 18044 df-minusg 18045 df-ghm 18294 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-rnghom 19396 df-staf 19545 df-srng 19546 df-lmod 19565 df-lmhm 19723 df-lvec 19804 df-sra 19873 df-rgmod 19874 df-phl 20698 |
This theorem is referenced by: cph2di 23738 |
Copyright terms: Public domain | W3C validator |