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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lkrss2N | Structured version Visualization version GIF version |
Description: Two functionals with kernels in a subset relationship. (Contributed by NM, 17-Feb-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lkrss2.s | ⊢ 𝑆 = (Scalar‘𝑊) |
lkrss2.r | ⊢ 𝑅 = (Base‘𝑆) |
lkrss2.f | ⊢ 𝐹 = (LFnl‘𝑊) |
lkrss2.k | ⊢ 𝐾 = (LKer‘𝑊) |
lkrss2.d | ⊢ 𝐷 = (LDual‘𝑊) |
lkrss2.t | ⊢ · = ( ·𝑠 ‘𝐷) |
lkrss2.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
lkrss2.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lkrss2.h | ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
Ref | Expression |
---|---|
lkrss2N | ⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspss 4044 | . . 3 ⊢ ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ∨ (𝐾‘𝐺) = (𝐾‘𝐻))) | |
2 | lkrss2.f | . . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑊) | |
3 | lkrss2.k | . . . . . . 7 ⊢ 𝐾 = (LKer‘𝑊) | |
4 | lkrss2.d | . . . . . . 7 ⊢ 𝐷 = (LDual‘𝑊) | |
5 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝐷) = (0g‘𝐷) | |
6 | lkrss2.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | lkrss2.g | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
8 | lkrss2.h | . . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ 𝐹) | |
9 | 2, 3, 4, 5, 6, 7, 8 | lkrpssN 37389 | . . . . . 6 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)))) |
10 | lveclmod 20439 | . . . . . . . . . . . 12 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) | |
11 | 6, 10 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ LMod) |
12 | lkrss2.s | . . . . . . . . . . . 12 ⊢ 𝑆 = (Scalar‘𝑊) | |
13 | lkrss2.r | . . . . . . . . . . . 12 ⊢ 𝑅 = (Base‘𝑆) | |
14 | eqid 2737 | . . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
15 | 12, 13, 14 | lmod0cl 20220 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
16 | 11, 15 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝑅) |
17 | 16 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → (0g‘𝑆) ∈ 𝑅) |
18 | simpr 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → 𝐻 = (0g‘𝐷)) | |
19 | lkrss2.t | . . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝐷) | |
20 | 2, 12, 14, 4, 19, 5, 11, 7 | ldual0vs 37386 | . . . . . . . . . . 11 ⊢ (𝜑 → ((0g‘𝑆) · 𝐺) = (0g‘𝐷)) |
21 | 20 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → ((0g‘𝑆) · 𝐺) = (0g‘𝐷)) |
22 | 18, 21 | eqtr4d 2780 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → 𝐻 = ((0g‘𝑆) · 𝐺)) |
23 | oveq1 7320 | . . . . . . . . . 10 ⊢ (𝑟 = (0g‘𝑆) → (𝑟 · 𝐺) = ((0g‘𝑆) · 𝐺)) | |
24 | 23 | rspceeqv 3584 | . . . . . . . . 9 ⊢ (((0g‘𝑆) ∈ 𝑅 ∧ 𝐻 = ((0g‘𝑆) · 𝐺)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
25 | 17, 22, 24 | syl2anc 584 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
26 | 25 | ex 413 | . . . . . . 7 ⊢ (𝜑 → (𝐻 = (0g‘𝐷) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
27 | 26 | adantld 491 | . . . . . 6 ⊢ (𝜑 → ((𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
28 | 9, 27 | sylbid 239 | . . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
29 | 28 | imp 407 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
30 | 6 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝑊 ∈ LVec) |
31 | 7 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐺 ∈ 𝐹) |
32 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐻 ∈ 𝐹) |
33 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → (𝐾‘𝐺) = (𝐾‘𝐻)) | |
34 | 12, 13, 2, 3, 4, 19, 30, 31, 32, 33 | eqlkr4 37391 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
35 | 29, 34 | jaodan 955 | . . 3 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ∨ (𝐾‘𝐺) = (𝐾‘𝐻))) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
36 | 1, 35 | sylan2b 594 | . 2 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
37 | 6 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑊 ∈ LVec) |
38 | 7 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐺 ∈ 𝐹) |
39 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅) | |
40 | 12, 13, 2, 3, 4, 19, 37, 38, 39 | lkrss 37394 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺))) |
41 | 40 | ex 413 | . . . . 5 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
42 | fveq2 6809 | . . . . . . 7 ⊢ (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐻) = (𝐾‘(𝑟 · 𝐺))) | |
43 | 42 | sseq2d 3962 | . . . . . 6 ⊢ (𝐻 = (𝑟 · 𝐺) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
44 | 43 | biimprcd 249 | . . . . 5 ⊢ ((𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)) → (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))) |
45 | 41, 44 | syl6 35 | . . . 4 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻)))) |
46 | 45 | rexlimdv 3147 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))) |
47 | 46 | imp 407 | . 2 ⊢ ((𝜑 ∧ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) |
48 | 36, 47 | impbida 798 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 = wceq 1540 ∈ wcel 2105 ≠ wne 2941 ∃wrex 3071 ⊆ wss 3896 ⊊ wpss 3897 ‘cfv 6463 (class class class)co 7313 Basecbs 16979 Scalarcsca 17032 ·𝑠 cvsca 17033 0gc0g 17217 LModclmod 20194 LVecclvec 20435 LFnlclfn 37283 LKerclk 37311 LDualcld 37349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5222 ax-sep 5236 ax-nul 5243 ax-pow 5301 ax-pr 5365 ax-un 7626 ax-cnex 10997 ax-resscn 10998 ax-1cn 10999 ax-icn 11000 ax-addcl 11001 ax-addrcl 11002 ax-mulcl 11003 ax-mulrcl 11004 ax-mulcom 11005 ax-addass 11006 ax-mulass 11007 ax-distr 11008 ax-i2m1 11009 ax-1ne0 11010 ax-1rid 11011 ax-rnegex 11012 ax-rrecex 11013 ax-cnre 11014 ax-pre-lttri 11015 ax-pre-lttrn 11016 ax-pre-ltadd 11017 ax-pre-mulgt0 11018 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3350 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3726 df-csb 3842 df-dif 3899 df-un 3901 df-in 3903 df-ss 3913 df-pss 3915 df-nul 4267 df-if 4470 df-pw 4545 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4849 df-int 4891 df-iun 4937 df-br 5086 df-opab 5148 df-mpt 5169 df-tr 5203 df-id 5505 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5560 df-we 5562 df-xp 5611 df-rel 5612 df-cnv 5613 df-co 5614 df-dm 5615 df-rn 5616 df-res 5617 df-ima 5618 df-pred 6222 df-ord 6289 df-on 6290 df-lim 6291 df-suc 6292 df-iota 6415 df-fun 6465 df-fn 6466 df-f 6467 df-f1 6468 df-fo 6469 df-f1o 6470 df-fv 6471 df-riota 7270 df-ov 7316 df-oprab 7317 df-mpo 7318 df-of 7571 df-om 7756 df-1st 7874 df-2nd 7875 df-tpos 8087 df-frecs 8142 df-wrecs 8173 df-recs 8247 df-rdg 8286 df-1o 8342 df-er 8544 df-map 8663 df-en 8780 df-dom 8781 df-sdom 8782 df-fin 8783 df-pnf 11081 df-mnf 11082 df-xr 11083 df-ltxr 11084 df-le 11085 df-sub 11277 df-neg 11278 df-nn 12044 df-2 12106 df-3 12107 df-4 12108 df-5 12109 df-6 12110 df-n0 12304 df-z 12390 df-uz 12653 df-fz 13310 df-struct 16915 df-sets 16932 df-slot 16950 df-ndx 16962 df-base 16980 df-ress 17009 df-plusg 17042 df-mulr 17043 df-sca 17045 df-vsca 17046 df-0g 17219 df-mgm 18393 df-sgrp 18442 df-mnd 18453 df-submnd 18498 df-grp 18647 df-minusg 18648 df-sbg 18649 df-subg 18819 df-cntz 18990 df-lsm 19308 df-cmn 19455 df-abl 19456 df-mgp 19788 df-ur 19805 df-ring 19852 df-oppr 19929 df-dvdsr 19950 df-unit 19951 df-invr 19981 df-drng 20064 df-lmod 20196 df-lss 20265 df-lsp 20305 df-lvec 20436 df-lshyp 37203 df-lfl 37284 df-lkr 37312 df-ldual 37350 |
This theorem is referenced by: lcfrvalsnN 39767 |
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