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 20757 | . . . . 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 19631 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉) |
11 | 5, 6, 7, 10 | syl3anc 1367 | . . 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 20766 | . . 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 1368 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷)))) |
17 | 12, 13, 8, 9, 14 | ipdi 20767 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
18 | 1, 2, 6, 7, 17 | syl13anc 1368 | . . 3 ⊢ (𝜑 → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
19 | 12, 13, 8, 9, 14 | ipdi 20767 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
20 | 1, 3, 6, 7, 19 | syl13anc 1368 | . . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
21 | 12 | phlsrng 20758 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
22 | srngring 19606 | . . . . . 6 ⊢ (𝐹 ∈ *-Ring → 𝐹 ∈ Ring) | |
23 | ringcmn 19314 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ CMnd) | |
24 | 1, 21, 22, 23 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ CMnd) |
25 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 12, 13, 8, 25 | ipcl 20760 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
27 | 1, 3, 6, 26 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
28 | 12, 13, 8, 25 | ipcl 20760 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
29 | 1, 3, 7, 28 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
30 | 25, 14 | cmncom 18906 | . . . . 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 1367 | . . . 4 ⊢ (𝜑 → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
32 | 20, 31 | eqtrd 2856 | . . 3 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
33 | 18, 32 | oveq12d 7160 | . 2 ⊢ (𝜑 → ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))) = (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
34 | 12, 13, 8, 25 | ipcl 20760 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
35 | 1, 2, 6, 34 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
36 | 12, 13, 8, 25 | ipcl 20760 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
37 | 1, 2, 7, 36 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
38 | 25, 14 | cmn4 18909 | . . 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 1375 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
40 | 16, 33, 39 | 3eqtrd 2860 | 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 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 Basecbs 16466 +gcplusg 16548 Scalarcsca 16551 ·𝑖cip 16553 CMndccmn 18889 Ringcrg 19280 *-Ringcsr 19598 LModclmod 19617 PreHilcphl 20751 |
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 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
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-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-tpos 7878 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-er 8275 df-map 8394 df-en 8496 df-dom 8497 df-sdom 8498 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-ndx 16469 df-slot 16470 df-base 16472 df-sets 16473 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-0g 16698 df-mgm 17835 df-sgrp 17884 df-mnd 17895 df-mhm 17939 df-grp 18089 df-minusg 18090 df-ghm 18339 df-cmn 18891 df-abl 18892 df-mgp 19223 df-ur 19235 df-ring 19282 df-oppr 19356 df-rnghom 19450 df-staf 19599 df-srng 19600 df-lmod 19619 df-lmhm 19777 df-lvec 19858 df-sra 19927 df-rgmod 19928 df-phl 20753 |
This theorem is referenced by: cph2di 23794 |
Copyright terms: Public domain | W3C validator |