![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ipsubdi | Structured version Visualization version GIF version |
Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ipsubdir.m | ⊢ − = (-g‘𝑊) |
ipsubdir.s | ⊢ 𝑆 = (-g‘𝐹) |
Ref | Expression |
---|---|
ipsubdi | ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ PreHil) | |
2 | simpr1 1193 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | |
3 | phllmod 21666 | . . . . . . . 8 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
4 | 3 | adantr 480 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ LMod) |
5 | lmodgrp 20882 | . . . . . . 7 ⊢ (𝑊 ∈ LMod → 𝑊 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑊 ∈ Grp) |
7 | simpr2 1194 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | |
8 | simpr3 1195 | . . . . . 6 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ 𝑉) | |
9 | phllmhm.v | . . . . . . 7 ⊢ 𝑉 = (Base‘𝑊) | |
10 | ipsubdir.m | . . . . . . 7 ⊢ − = (-g‘𝑊) | |
11 | 9, 10 | grpsubcl 19051 | . . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐵 − 𝐶) ∈ 𝑉) |
12 | 6, 7, 8, 11 | syl3anc 1370 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 − 𝐶) ∈ 𝑉) |
13 | phlsrng.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
14 | phllmhm.h | . . . . . 6 ⊢ , = (·𝑖‘𝑊) | |
15 | eqid 2735 | . . . . . 6 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
16 | eqid 2735 | . . . . . 6 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
17 | 13, 14, 9, 15, 16 | ipdi 21676 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ (𝐵 − 𝐶) ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , ((𝐵 − 𝐶)(+g‘𝑊)𝐶)) = ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶))) |
18 | 1, 2, 12, 8, 17 | syl13anc 1371 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , ((𝐵 − 𝐶)(+g‘𝑊)𝐶)) = ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶))) |
19 | 9, 15, 10 | grpnpcan 19063 | . . . . . 6 ⊢ ((𝑊 ∈ Grp ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → ((𝐵 − 𝐶)(+g‘𝑊)𝐶) = 𝐵) |
20 | 6, 7, 8, 19 | syl3anc 1370 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐵 − 𝐶)(+g‘𝑊)𝐶) = 𝐵) |
21 | 20 | oveq2d 7447 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , ((𝐵 − 𝐶)(+g‘𝑊)𝐶)) = (𝐴 , 𝐵)) |
22 | 18, 21 | eqtr3d 2777 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶)) = (𝐴 , 𝐵)) |
23 | 13 | lmodfgrp 20884 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Grp) |
24 | 4, 23 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝐹 ∈ Grp) |
25 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
26 | 13, 14, 9, 25 | ipcl 21669 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ (Base‘𝐹)) |
27 | 1, 2, 7, 26 | syl3anc 1370 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , 𝐵) ∈ (Base‘𝐹)) |
28 | 13, 14, 9, 25 | ipcl 21669 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
29 | 1, 2, 8, 28 | syl3anc 1370 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , 𝐶) ∈ (Base‘𝐹)) |
30 | 13, 14, 9, 25 | ipcl 21669 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ (𝐵 − 𝐶) ∈ 𝑉) → (𝐴 , (𝐵 − 𝐶)) ∈ (Base‘𝐹)) |
31 | 1, 2, 12, 30 | syl3anc 1370 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) ∈ (Base‘𝐹)) |
32 | ipsubdir.s | . . . . 5 ⊢ 𝑆 = (-g‘𝐹) | |
33 | 25, 16, 32 | grpsubadd 19059 | . . . 4 ⊢ ((𝐹 ∈ Grp ∧ ((𝐴 , 𝐵) ∈ (Base‘𝐹) ∧ (𝐴 , 𝐶) ∈ (Base‘𝐹) ∧ (𝐴 , (𝐵 − 𝐶)) ∈ (Base‘𝐹))) → (((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)) = (𝐴 , (𝐵 − 𝐶)) ↔ ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶)) = (𝐴 , 𝐵))) |
34 | 24, 27, 29, 31, 33 | syl13anc 1371 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)) = (𝐴 , (𝐵 − 𝐶)) ↔ ((𝐴 , (𝐵 − 𝐶))(+g‘𝐹)(𝐴 , 𝐶)) = (𝐴 , 𝐵))) |
35 | 22, 34 | mpbird 257 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)) = (𝐴 , (𝐵 − 𝐶))) |
36 | 35 | eqcomd 2741 | 1 ⊢ ((𝑊 ∈ PreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Scalarcsca 17301 ·𝑖cip 17303 Grpcgrp 18964 -gcsg 18966 LModclmod 20875 PreHilcphl 21660 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-grp 18967 df-minusg 18968 df-sbg 18969 df-ghm 19244 df-mgp 20153 df-ur 20200 df-ring 20253 df-oppr 20351 df-rhm 20489 df-staf 20857 df-srng 20858 df-lmod 20877 df-lmhm 21039 df-lvec 21120 df-sra 21190 df-rgmod 21191 df-phl 21662 |
This theorem is referenced by: ip2subdi 21680 ip2eq 21689 cphsubdi 25257 |
Copyright terms: Public domain | W3C validator |