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Mirrors > Home > MPE Home > Th. List > nlmdsdi | Structured version Visualization version GIF version |
Description: Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
nlmdsdi.v | ⊢ 𝑉 = (Base‘𝑊) |
nlmdsdi.s | ⊢ · = ( ·𝑠 ‘𝑊) |
nlmdsdi.f | ⊢ 𝐹 = (Scalar‘𝑊) |
nlmdsdi.k | ⊢ 𝐾 = (Base‘𝐹) |
nlmdsdi.d | ⊢ 𝐷 = (dist‘𝑊) |
nlmdsdi.a | ⊢ 𝐴 = (norm‘𝐹) |
Ref | Expression |
---|---|
nlmdsdi | ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝐴‘𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ NrmMod) | |
2 | simpr1 1195 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑋 ∈ 𝐾) | |
3 | nlmngp 23969 | . . . . . . 7 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp) | |
4 | 3 | adantr 482 | . . . . . 6 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ NrmGrp) |
5 | ngpgrp 23883 | . . . . . 6 ⊢ (𝑊 ∈ NrmGrp → 𝑊 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ Grp) |
7 | simpr2 1196 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑌 ∈ 𝑉) | |
8 | simpr3 1197 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑍 ∈ 𝑉) | |
9 | nlmdsdi.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
10 | eqid 2738 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
11 | 9, 10 | grpsubcl 18762 | . . . . 5 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌(-g‘𝑊)𝑍) ∈ 𝑉) |
12 | 6, 7, 8, 11 | syl3anc 1372 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → (𝑌(-g‘𝑊)𝑍) ∈ 𝑉) |
13 | eqid 2738 | . . . . 5 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
14 | nlmdsdi.s | . . . . 5 ⊢ · = ( ·𝑠 ‘𝑊) | |
15 | nlmdsdi.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
16 | nlmdsdi.k | . . . . 5 ⊢ 𝐾 = (Base‘𝐹) | |
17 | nlmdsdi.a | . . . . 5 ⊢ 𝐴 = (norm‘𝐹) | |
18 | 9, 13, 14, 15, 16, 17 | nmvs 23968 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ (𝑌(-g‘𝑊)𝑍) ∈ 𝑉) → ((norm‘𝑊)‘(𝑋 · (𝑌(-g‘𝑊)𝑍))) = ((𝐴‘𝑋) · ((norm‘𝑊)‘(𝑌(-g‘𝑊)𝑍)))) |
19 | 1, 2, 12, 18 | syl3anc 1372 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((norm‘𝑊)‘(𝑋 · (𝑌(-g‘𝑊)𝑍))) = ((𝐴‘𝑋) · ((norm‘𝑊)‘(𝑌(-g‘𝑊)𝑍)))) |
20 | nlmlmod 23970 | . . . . . 6 ⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod) | |
21 | 20 | adantr 482 | . . . . 5 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → 𝑊 ∈ LMod) |
22 | 9, 14, 15, 16, 10, 21, 2, 7, 8 | lmodsubdi 20308 | . . . 4 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → (𝑋 · (𝑌(-g‘𝑊)𝑍)) = ((𝑋 · 𝑌)(-g‘𝑊)(𝑋 · 𝑍))) |
23 | 22 | fveq2d 6842 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((norm‘𝑊)‘(𝑋 · (𝑌(-g‘𝑊)𝑍))) = ((norm‘𝑊)‘((𝑋 · 𝑌)(-g‘𝑊)(𝑋 · 𝑍)))) |
24 | 19, 23 | eqtr3d 2780 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝐴‘𝑋) · ((norm‘𝑊)‘(𝑌(-g‘𝑊)𝑍))) = ((norm‘𝑊)‘((𝑋 · 𝑌)(-g‘𝑊)(𝑋 · 𝑍)))) |
25 | nlmdsdi.d | . . . . 5 ⊢ 𝐷 = (dist‘𝑊) | |
26 | 13, 9, 10, 25 | ngpds 23888 | . . . 4 ⊢ ((𝑊 ∈ NrmGrp ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌𝐷𝑍) = ((norm‘𝑊)‘(𝑌(-g‘𝑊)𝑍))) |
27 | 4, 7, 8, 26 | syl3anc 1372 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → (𝑌𝐷𝑍) = ((norm‘𝑊)‘(𝑌(-g‘𝑊)𝑍))) |
28 | 27 | oveq2d 7366 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝐴‘𝑋) · (𝑌𝐷𝑍)) = ((𝐴‘𝑋) · ((norm‘𝑊)‘(𝑌(-g‘𝑊)𝑍)))) |
29 | 9, 15, 14, 16 | lmodvscl 20268 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑋 · 𝑌) ∈ 𝑉) |
30 | 21, 2, 7, 29 | syl3anc 1372 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → (𝑋 · 𝑌) ∈ 𝑉) |
31 | 9, 15, 14, 16 | lmodvscl 20268 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉) → (𝑋 · 𝑍) ∈ 𝑉) |
32 | 21, 2, 8, 31 | syl3anc 1372 | . . 3 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → (𝑋 · 𝑍) ∈ 𝑉) |
33 | 13, 9, 10, 25 | ngpds 23888 | . . 3 ⊢ ((𝑊 ∈ NrmGrp ∧ (𝑋 · 𝑌) ∈ 𝑉 ∧ (𝑋 · 𝑍) ∈ 𝑉) → ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍)) = ((norm‘𝑊)‘((𝑋 · 𝑌)(-g‘𝑊)(𝑋 · 𝑍)))) |
34 | 4, 30, 32, 33 | syl3anc 1372 | . 2 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍)) = ((norm‘𝑊)‘((𝑋 · 𝑌)(-g‘𝑊)(𝑋 · 𝑍)))) |
35 | 24, 28, 34 | 3eqtr4d 2788 | 1 ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝐴‘𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ‘cfv 6492 (class class class)co 7350 · cmul 10990 Basecbs 17019 Scalarcsca 17072 ·𝑠 cvsca 17073 distcds 17078 Grpcgrp 18684 -gcsg 18686 LModclmod 20251 normcnm 23860 NrmGrpcngp 23861 NrmModcnlm 23864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-om 7794 df-1st 7912 df-2nd 7913 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-er 8582 df-map 8701 df-en 8818 df-dom 8819 df-sdom 8820 df-sup 9312 df-inf 9313 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-n0 12348 df-z 12434 df-uz 12698 df-q 12804 df-rp 12846 df-xneg 12963 df-xadd 12964 df-xmul 12965 df-sets 16972 df-slot 16990 df-ndx 17002 df-base 17020 df-plusg 17082 df-0g 17259 df-topgen 17261 df-mgm 18433 df-sgrp 18482 df-mnd 18493 df-grp 18687 df-minusg 18688 df-sbg 18689 df-mgp 19832 df-ur 19849 df-ring 19896 df-lmod 20253 df-psmet 20717 df-xmet 20718 df-met 20719 df-bl 20720 df-mopn 20721 df-top 22171 df-topon 22188 df-topsp 22210 df-bases 22224 df-xms 23601 df-ms 23602 df-nm 23866 df-ngp 23867 df-nlm 23870 |
This theorem is referenced by: nlmvscnlem2 23977 |
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