Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrsc | Structured version Visualization version GIF version |
Description: The kernel of a nonzero scalar product of a functional equals the kernel of the functional. (Contributed by NM, 9-Oct-2014.) |
Ref | Expression |
---|---|
lkrsc.v | ⊢ 𝑉 = (Base‘𝑊) |
lkrsc.d | ⊢ 𝐷 = (Scalar‘𝑊) |
lkrsc.k | ⊢ 𝐾 = (Base‘𝐷) |
lkrsc.t | ⊢ · = (.r‘𝐷) |
lkrsc.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrsc.l | ⊢ 𝐿 = (LKer‘𝑊) |
lkrsc.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrsc.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lkrsc.r | ⊢ (𝜑 → 𝑅 ∈ 𝐾) |
lkrsc.o | ⊢ 0 = (0g‘𝐷) |
lkrsc.e | ⊢ (𝜑 → 𝑅 ≠ 0 ) |
Ref | Expression |
---|---|
lkrsc | ⊢ (𝜑 → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lkrsc.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊) | |
2 | fvex 6113 | . . . . . . . . 9 ⊢ (Base‘𝑊) ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . . . . . . 8 ⊢ 𝑉 ∈ V |
4 | 3 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑉 ∈ V) |
5 | lkrsc.r | . . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ 𝐾) | |
6 | lkrsc.w | . . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lkrsc.g | . . . . . . . . 9 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
8 | lkrsc.d | . . . . . . . . . 10 ⊢ 𝐷 = (Scalar‘𝑊) | |
9 | lkrsc.k | . . . . . . . . . 10 ⊢ 𝐾 = (Base‘𝐷) | |
10 | lkrsc.f | . . . . . . . . . 10 ⊢ 𝐹 = (LFnl‘𝑊) | |
11 | 8, 9, 1, 10 | lflf 33368 | . . . . . . . . 9 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → 𝐺:𝑉⟶𝐾) |
12 | 6, 7, 11 | syl2anc 691 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑉⟶𝐾) |
13 | ffn 5958 | . . . . . . . 8 ⊢ (𝐺:𝑉⟶𝐾 → 𝐺 Fn 𝑉) | |
14 | 12, 13 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐺 Fn 𝑉) |
15 | eqidd 2611 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) = (𝐺‘𝑣)) | |
16 | 4, 5, 14, 15 | ofc2 6819 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → ((𝐺 ∘𝑓 · (𝑉 × {𝑅}))‘𝑣) = ((𝐺‘𝑣) · 𝑅)) |
17 | 16 | eqeq1d 2612 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘𝑓 · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ ((𝐺‘𝑣) · 𝑅) = 0 )) |
18 | lkrsc.o | . . . . . 6 ⊢ 0 = (0g‘𝐷) | |
19 | lkrsc.t | . . . . . 6 ⊢ · = (.r‘𝐷) | |
20 | 8 | lvecdrng 18926 | . . . . . . . 8 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ DivRing) |
21 | 6, 20 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ DivRing) |
22 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐷 ∈ DivRing) |
23 | 6 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑊 ∈ LVec) |
24 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝐺 ∈ 𝐹) |
25 | simpr 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑣 ∈ 𝑉) | |
26 | 8, 9, 1, 10 | lflcl 33369 | . . . . . . 7 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
27 | 23, 24, 25, 26 | syl3anc 1318 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (𝐺‘𝑣) ∈ 𝐾) |
28 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ∈ 𝐾) |
29 | lkrsc.e | . . . . . . 7 ⊢ (𝜑 → 𝑅 ≠ 0 ) | |
30 | 29 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → 𝑅 ≠ 0 ) |
31 | 9, 18, 19, 22, 27, 28, 30 | drngmuleq0 18593 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺‘𝑣) · 𝑅) = 0 ↔ (𝐺‘𝑣) = 0 )) |
32 | 17, 31 | bitrd 267 | . . . 4 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑉) → (((𝐺 ∘𝑓 · (𝑉 × {𝑅}))‘𝑣) = 0 ↔ (𝐺‘𝑣) = 0 )) |
33 | 32 | pm5.32da 671 | . . 3 ⊢ (𝜑 → ((𝑣 ∈ 𝑉 ∧ ((𝐺 ∘𝑓 · (𝑉 × {𝑅}))‘𝑣) = 0 ) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
34 | lveclmod 18927 | . . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
35 | 6, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) |
36 | 1, 8, 9, 19, 10, 35, 7, 5 | lflvscl 33382 | . . . 4 ⊢ (𝜑 → (𝐺 ∘𝑓 · (𝑉 × {𝑅})) ∈ 𝐹) |
37 | lkrsc.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑊) | |
38 | 1, 8, 18, 10, 37 | ellkr 33394 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∘𝑓 · (𝑉 × {𝑅})) ∈ 𝐹) → (𝑣 ∈ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘𝑓 · (𝑉 × {𝑅}))‘𝑣) = 0 ))) |
39 | 6, 36, 38 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) ↔ (𝑣 ∈ 𝑉 ∧ ((𝐺 ∘𝑓 · (𝑉 × {𝑅}))‘𝑣) = 0 ))) |
40 | 1, 8, 18, 10, 37 | ellkr 33394 | . . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
41 | 6, 7, 40 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘𝐺) ↔ (𝑣 ∈ 𝑉 ∧ (𝐺‘𝑣) = 0 ))) |
42 | 33, 39, 41 | 3bitr4d 299 | . 2 ⊢ (𝜑 → (𝑣 ∈ (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) ↔ 𝑣 ∈ (𝐿‘𝐺))) |
43 | 42 | eqrdv 2608 | 1 ⊢ (𝜑 → (𝐿‘(𝐺 ∘𝑓 · (𝑉 × {𝑅}))) = (𝐿‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 {csn 4125 × cxp 5036 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ∘𝑓 cof 6793 Basecbs 15695 .rcmulr 15769 Scalarcsca 15771 0gc0g 15923 DivRingcdr 18570 LModclmod 18686 LVecclvec 18923 LFnlclfn 33362 LKerclk 33390 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-mgp 18313 df-ur 18325 df-ring 18372 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-drng 18572 df-lmod 18688 df-lvec 18924 df-lfl 33363 df-lkr 33391 |
This theorem is referenced by: lkrscss 33403 ldualkrsc 33472 |
Copyright terms: Public domain | W3C validator |