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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 4059 | . . 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 2736 | . . . . . . 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 37625 | . . . . . 6 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)))) |
10 | lveclmod 20567 | . . . . . . . . . . . 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 2736 | . . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
15 | 12, 13, 14 | lmod0cl 20348 | . . . . . . . . . . 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 37622 | . . . . . . . . . . 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 2779 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → 𝐻 = ((0g‘𝑆) · 𝐺)) |
23 | oveq1 7364 | . . . . . . . . . 10 ⊢ (𝑟 = (0g‘𝑆) → (𝑟 · 𝐺) = ((0g‘𝑆) · 𝐺)) | |
24 | 23 | rspceeqv 3595 | . . . . . . . . 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 37627 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
35 | 29, 34 | jaodan 956 | . . 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 37630 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺))) |
41 | 40 | ex 413 | . . . . 5 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
42 | fveq2 6842 | . . . . . . 7 ⊢ (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐻) = (𝐾‘(𝑟 · 𝐺))) | |
43 | 42 | sseq2d 3976 | . . . . . 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 3150 | . . 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 799 | 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 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 ⊆ wss 3910 ⊊ wpss 3911 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 Scalarcsca 17136 ·𝑠 cvsca 17137 0gc0g 17321 LModclmod 20322 LVecclvec 20563 LFnlclfn 37519 LKerclk 37547 LDualcld 37585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lshyp 37439 df-lfl 37520 df-lkr 37548 df-ldual 37586 |
This theorem is referenced by: lcfrvalsnN 40004 |
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