Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lflmul | Structured version Visualization version GIF version |
Description: Property of a linear functional. (lnfnmuli 29815 analog.) (Contributed by NM, 16-Apr-2014.) |
Ref | Expression |
---|---|
lflmul.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lflmul.k | ⊢ 𝐾 = (Base‘𝐷) |
lflmul.t | ⊢ × = (.r‘𝐷) |
lflmul.v | ⊢ 𝑉 = (Base‘𝑊) |
lflmul.s | ⊢ · = ( ·𝑠 ‘𝑊) |
lflmul.f | ⊢ 𝐹 = (LFnl‘𝑊) |
Ref | Expression |
---|---|
lflmul | ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘(𝑅 · 𝑋)) = (𝑅 × (𝐺‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → 𝑊 ∈ LMod) | |
2 | simp2 1133 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → 𝐺 ∈ 𝐹) | |
3 | simp3l 1197 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → 𝑅 ∈ 𝐾) | |
4 | simp3r 1198 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → 𝑋 ∈ 𝑉) | |
5 | lflmul.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
6 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
7 | 5, 6 | lmod0vcl 19657 | . . . 4 ⊢ (𝑊 ∈ LMod → (0g‘𝑊) ∈ 𝑉) |
8 | 7 | 3ad2ant1 1129 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (0g‘𝑊) ∈ 𝑉) |
9 | eqid 2821 | . . . 4 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
10 | lflmul.d | . . . 4 ⊢ 𝐷 = (Scalar‘𝑊) | |
11 | lflmul.s | . . . 4 ⊢ · = ( ·𝑠 ‘𝑊) | |
12 | lflmul.k | . . . 4 ⊢ 𝐾 = (Base‘𝐷) | |
13 | eqid 2821 | . . . 4 ⊢ (+g‘𝐷) = (+g‘𝐷) | |
14 | lflmul.t | . . . 4 ⊢ × = (.r‘𝐷) | |
15 | lflmul.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑊) | |
16 | 5, 9, 10, 11, 12, 13, 14, 15 | lfli 36191 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ (0g‘𝑊) ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋)(+g‘𝑊)(0g‘𝑊))) = ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(𝐺‘(0g‘𝑊)))) |
17 | 1, 2, 3, 4, 8, 16 | syl113anc 1378 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋)(+g‘𝑊)(0g‘𝑊))) = ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(𝐺‘(0g‘𝑊)))) |
18 | 5, 10, 11, 12 | lmodvscl 19645 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑅 · 𝑋) ∈ 𝑉) |
19 | 1, 3, 4, 18 | syl3anc 1367 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · 𝑋) ∈ 𝑉) |
20 | 5, 9, 6 | lmod0vrid 19659 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ (𝑅 · 𝑋) ∈ 𝑉) → ((𝑅 · 𝑋)(+g‘𝑊)(0g‘𝑊)) = (𝑅 · 𝑋)) |
21 | 1, 19, 20 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 · 𝑋)(+g‘𝑊)(0g‘𝑊)) = (𝑅 · 𝑋)) |
22 | 21 | fveq2d 6669 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘((𝑅 · 𝑋)(+g‘𝑊)(0g‘𝑊))) = (𝐺‘(𝑅 · 𝑋))) |
23 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
24 | 10, 23, 6, 15 | lfl0 36195 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝐺‘(0g‘𝑊)) = (0g‘𝐷)) |
25 | 24 | 3adant3 1128 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘(0g‘𝑊)) = (0g‘𝐷)) |
26 | 25 | oveq2d 7166 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(𝐺‘(0g‘𝑊))) = ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(0g‘𝐷))) |
27 | 10 | lmodfgrp 19637 | . . . . 5 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Grp) |
28 | 27 | 3ad2ant1 1129 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → 𝐷 ∈ Grp) |
29 | 10, 12, 5, 15 | lflcl 36194 | . . . . . 6 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ 𝐾) |
30 | 29 | 3adant3l 1176 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘𝑋) ∈ 𝐾) |
31 | 10, 12, 14 | lmodmcl 19640 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ (𝐺‘𝑋) ∈ 𝐾) → (𝑅 × (𝐺‘𝑋)) ∈ 𝐾) |
32 | 1, 3, 30, 31 | syl3anc 1367 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑅 × (𝐺‘𝑋)) ∈ 𝐾) |
33 | 12, 13, 23 | grprid 18128 | . . . 4 ⊢ ((𝐷 ∈ Grp ∧ (𝑅 × (𝐺‘𝑋)) ∈ 𝐾) → ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(0g‘𝐷)) = (𝑅 × (𝐺‘𝑋))) |
34 | 28, 32, 33 | syl2anc 586 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(0g‘𝐷)) = (𝑅 × (𝐺‘𝑋))) |
35 | 26, 34 | eqtrd 2856 | . 2 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑅 × (𝐺‘𝑋))(+g‘𝐷)(𝐺‘(0g‘𝑊))) = (𝑅 × (𝐺‘𝑋))) |
36 | 17, 22, 35 | 3eqtr3d 2864 | 1 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝐺‘(𝑅 · 𝑋)) = (𝑅 × (𝐺‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 Scalarcsca 16562 ·𝑠 cvsca 16563 0gc0g 16707 Grpcgrp 18097 LModclmod 19628 LFnlclfn 36187 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mgp 19234 df-ur 19246 df-ring 19293 df-lmod 19630 df-lfl 36188 |
This theorem is referenced by: lfl1 36200 lfladdcl 36201 eqlkr 36229 lkrlsp 36232 dochkr1 38608 dochkr1OLDN 38609 lcfl7lem 38629 lclkrlem2m 38649 hdmaplnm1 39039 |
Copyright terms: Public domain | W3C validator |