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 20319 | . . . . 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 19641 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐶 + 𝐷) ∈ 𝑉) |
11 | 5, 6, 7, 10 | syl3anc 1368 | . . 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 20328 | . . 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 1369 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷)))) |
17 | 12, 13, 8, 9, 14 | ipdi 20329 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
18 | 1, 2, 6, 7, 17 | syl13anc 1369 | . . 3 ⊢ (𝜑 → (𝐴 , (𝐶 + 𝐷)) = ((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷))) |
19 | 12, 13, 8, 9, 14 | ipdi 20329 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
20 | 1, 3, 6, 7, 19 | syl13anc 1369 | . . . 4 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷))) |
21 | 12 | phlsrng 20320 | . . . . . 6 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
22 | srngring 19616 | . . . . . 6 ⊢ (𝐹 ∈ *-Ring → 𝐹 ∈ Ring) | |
23 | ringcmn 19327 | . . . . . 6 ⊢ (𝐹 ∈ Ring → 𝐹 ∈ CMnd) | |
24 | 1, 21, 22, 23 | 4syl 19 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ CMnd) |
25 | eqid 2798 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 12, 13, 8, 25 | ipcl 20322 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
27 | 1, 3, 6, 26 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐶) ∈ (Base‘𝐹)) |
28 | 12, 13, 8, 25 | ipcl 20322 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
29 | 1, 3, 7, 28 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → (𝐵 , 𝐷) ∈ (Base‘𝐹)) |
30 | 25, 14 | cmncom 18915 | . . . . 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 1368 | . . . 4 ⊢ (𝜑 → ((𝐵 , 𝐶) ⨣ (𝐵 , 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
32 | 20, 31 | eqtrd 2833 | . . 3 ⊢ (𝜑 → (𝐵 , (𝐶 + 𝐷)) = ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) |
33 | 18, 32 | oveq12d 7153 | . 2 ⊢ (𝜑 → ((𝐴 , (𝐶 + 𝐷)) ⨣ (𝐵 , (𝐶 + 𝐷))) = (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
34 | 12, 13, 8, 25 | ipcl 20322 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
35 | 1, 2, 6, 34 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
36 | 12, 13, 8, 25 | ipcl 20322 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉) → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
37 | 1, 2, 7, 36 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐴 , 𝐷) ∈ (Base‘𝐹)) |
38 | 25, 14 | cmn4 18918 | . . 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 1376 | . 2 ⊢ (𝜑 → (((𝐴 , 𝐶) ⨣ (𝐴 , 𝐷)) ⨣ ((𝐵 , 𝐷) ⨣ (𝐵 , 𝐶))) = (((𝐴 , 𝐶) ⨣ (𝐵 , 𝐷)) ⨣ ((𝐴 , 𝐷) ⨣ (𝐵 , 𝐶)))) |
40 | 16, 33, 39 | 3eqtrd 2837 | 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 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Scalarcsca 16560 ·𝑖cip 16562 CMndccmn 18898 Ringcrg 19290 *-Ringcsr 19608 LModclmod 19627 PreHilcphl 20313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-grp 18098 df-minusg 18099 df-ghm 18348 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-rnghom 19463 df-staf 19609 df-srng 19610 df-lmod 19629 df-lmhm 19787 df-lvec 19868 df-sra 19937 df-rgmod 19938 df-phl 20315 |
This theorem is referenced by: cph2di 23812 |
Copyright terms: Public domain | W3C validator |