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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 3932 | . . 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 2825 | . . . . . . 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 35238 | . . . . . 6 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)))) |
10 | lveclmod 19465 | . . . . . . . . . . . 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 2825 | . . . . . . . . . . . 12 ⊢ (0g‘𝑆) = (0g‘𝑆) | |
15 | 12, 13, 14 | lmod0cl 19245 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
16 | 11, 15 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝑅) |
17 | 16 | adantr 474 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → (0g‘𝑆) ∈ 𝑅) |
18 | simpr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → 𝐻 = (0g‘𝐷)) | |
19 | lkrss2.t | . . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝐷) | |
20 | 2, 12, 14, 4, 19, 5, 11, 7 | ldual0vs 35235 | . . . . . . . . . . 11 ⊢ (𝜑 → ((0g‘𝑆) · 𝐺) = (0g‘𝐷)) |
21 | 20 | adantr 474 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → ((0g‘𝑆) · 𝐺) = (0g‘𝐷)) |
22 | 18, 21 | eqtr4d 2864 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → 𝐻 = ((0g‘𝑆) · 𝐺)) |
23 | oveq1 6912 | . . . . . . . . . 10 ⊢ (𝑟 = (0g‘𝑆) → (𝑟 · 𝐺) = ((0g‘𝑆) · 𝐺)) | |
24 | 23 | rspceeqv 3544 | . . . . . . . . 9 ⊢ (((0g‘𝑆) ∈ 𝑅 ∧ 𝐻 = ((0g‘𝑆) · 𝐺)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
25 | 17, 22, 24 | syl2anc 581 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
26 | 25 | ex 403 | . . . . . . 7 ⊢ (𝜑 → (𝐻 = (0g‘𝐷) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
27 | 26 | adantld 486 | . . . . . 6 ⊢ (𝜑 → ((𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
28 | 9, 27 | sylbid 232 | . . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
29 | 28 | imp 397 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
30 | 6 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝑊 ∈ LVec) |
31 | 7 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐺 ∈ 𝐹) |
32 | 8 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐻 ∈ 𝐹) |
33 | simpr 479 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → (𝐾‘𝐺) = (𝐾‘𝐻)) | |
34 | 12, 13, 2, 3, 4, 19, 30, 31, 32, 33 | eqlkr4 35240 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
35 | 29, 34 | jaodan 987 | . . 3 ⊢ ((𝜑 ∧ ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ∨ (𝐾‘𝐺) = (𝐾‘𝐻))) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
36 | 1, 35 | sylan2b 589 | . 2 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
37 | 6 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑊 ∈ LVec) |
38 | 7 | adantr 474 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐺 ∈ 𝐹) |
39 | simpr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅) | |
40 | 12, 13, 2, 3, 4, 19, 37, 38, 39 | lkrss 35243 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺))) |
41 | 40 | ex 403 | . . . . 5 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
42 | fveq2 6433 | . . . . . . 7 ⊢ (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐻) = (𝐾‘(𝑟 · 𝐺))) | |
43 | 42 | sseq2d 3858 | . . . . . 6 ⊢ (𝐻 = (𝑟 · 𝐺) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
44 | 43 | biimprcd 242 | . . . . 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 3239 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))) |
47 | 46 | imp 397 | . 2 ⊢ ((𝜑 ∧ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) |
48 | 36, 47 | impbida 837 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∨ wo 880 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∃wrex 3118 ⊆ wss 3798 ⊊ wpss 3799 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 Scalarcsca 16308 ·𝑠 cvsca 16309 0gc0g 16453 LModclmod 19219 LVecclvec 19461 LFnlclfn 35132 LKerclk 35160 LDualcld 35198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lshyp 35052 df-lfl 35133 df-lkr 35161 df-ldual 35199 |
This theorem is referenced by: lcfrvalsnN 37616 |
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