| Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
| 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 4102 | . . 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 39164 | . . . . . 6 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) ↔ (𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)))) |
| 10 | lveclmod 21105 | . . . . . . . . . . . 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 20886 | . . . . . . . . . . 11 ⊢ (𝑊 ∈ LMod → (0g‘𝑆) ∈ 𝑅) |
| 16 | 11, 15 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝑅) |
| 17 | 16 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → (0g‘𝑆) ∈ 𝑅) |
| 18 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐻 = (0g‘𝐷)) → 𝐻 = (0g‘𝐷)) | |
| 19 | lkrss2.t | . . . . . . . . . . . 12 ⊢ · = ( ·𝑠 ‘𝐷) | |
| 20 | 2, 12, 14, 4, 19, 5, 11, 7 | ldual0vs 39161 | . . . . . . . . . . 11 ⊢ (𝜑 → ((0g‘𝑆) · 𝐺) = (0g‘𝐷)) |
| 21 | 20 | adantr 480 | . . . . . . . . . 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 7438 | . . . . . . . . . 10 ⊢ (𝑟 = (0g‘𝑆) → (𝑟 · 𝐺) = ((0g‘𝑆) · 𝐺)) | |
| 24 | 23 | rspceeqv 3645 | . . . . . . . . 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 412 | . . . . . . 7 ⊢ (𝜑 → (𝐻 = (0g‘𝐷) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
| 27 | 26 | adantld 490 | . . . . . 6 ⊢ (𝜑 → ((𝐺 ≠ (0g‘𝐷) ∧ 𝐻 = (0g‘𝐷)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
| 28 | 9, 27 | sylbid 240 | . . . . 5 ⊢ (𝜑 → ((𝐾‘𝐺) ⊊ (𝐾‘𝐻) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
| 29 | 28 | imp 406 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) ⊊ (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
| 30 | 6 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝑊 ∈ LVec) |
| 31 | 7 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐺 ∈ 𝐹) |
| 32 | 8 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → 𝐻 ∈ 𝐹) |
| 33 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → (𝐾‘𝐺) = (𝐾‘𝐻)) | |
| 34 | 12, 13, 2, 3, 4, 19, 30, 31, 32, 33 | eqlkr4 39166 | . . . 4 ⊢ ((𝜑 ∧ (𝐾‘𝐺) = (𝐾‘𝐻)) → ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) |
| 35 | 29, 34 | jaodan 960 | . . 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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑊 ∈ LVec) |
| 38 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝐺 ∈ 𝐹) |
| 39 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ 𝑅) | |
| 40 | 12, 13, 2, 3, 4, 19, 37, 38, 39 | lkrss 39169 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑟 ∈ 𝑅) → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺))) |
| 41 | 40 | ex 412 | . . . . 5 ⊢ (𝜑 → (𝑟 ∈ 𝑅 → (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
| 42 | fveq2 6906 | . . . . . . 7 ⊢ (𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐻) = (𝐾‘(𝑟 · 𝐺))) | |
| 43 | 42 | sseq2d 4016 | . . . . . 6 ⊢ (𝐻 = (𝑟 · 𝐺) → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ (𝐾‘𝐺) ⊆ (𝐾‘(𝑟 · 𝐺)))) |
| 44 | 43 | biimprcd 250 | . . . . 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 3153 | . . 3 ⊢ (𝜑 → (∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻))) |
| 47 | 46 | imp 406 | . 2 ⊢ ((𝜑 ∧ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺)) → (𝐾‘𝐺) ⊆ (𝐾‘𝐻)) |
| 48 | 36, 47 | impbida 801 | 1 ⊢ (𝜑 → ((𝐾‘𝐺) ⊆ (𝐾‘𝐻) ↔ ∃𝑟 ∈ 𝑅 𝐻 = (𝑟 · 𝐺))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3951 ⊊ wpss 3952 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 LModclmod 20858 LVecclvec 21101 LFnlclfn 39058 LKerclk 39086 LDualcld 39124 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-sbg 18956 df-subg 19141 df-cntz 19335 df-lsm 19654 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-nzr 20513 df-rlreg 20694 df-domn 20695 df-drng 20731 df-lmod 20860 df-lss 20930 df-lsp 20970 df-lvec 21102 df-lshyp 38978 df-lfl 39059 df-lkr 39087 df-ldual 39125 |
| This theorem is referenced by: lcfrvalsnN 41543 |
| Copyright terms: Public domain | W3C validator |