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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 37741. (Contributed by NM, 27-Feb-2015.) |
Ref | Expression |
---|---|
lcfrlem1.v | ⊢ 𝑉 = (Base‘𝑈) |
lcfrlem1.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lcfrlem1.q | ⊢ × = (.r‘𝑆) |
lcfrlem1.z | ⊢ 0 = (0g‘𝑆) |
lcfrlem1.i | ⊢ 𝐼 = (invr‘𝑆) |
lcfrlem1.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lcfrlem1.d | ⊢ 𝐷 = (LDual‘𝑈) |
lcfrlem1.t | ⊢ · = ( ·𝑠 ‘𝐷) |
lcfrlem1.m | ⊢ − = (-g‘𝐷) |
lcfrlem1.u | ⊢ (𝜑 → 𝑈 ∈ LVec) |
lcfrlem1.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lcfrlem1.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lcfrlem1.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
lcfrlem1.n | ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) |
lcfrlem1.h | ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) |
lcfrlem2.l | ⊢ 𝐿 = (LKer‘𝑈) |
Ref | Expression |
---|---|
lcfrlem3 | ⊢ (𝜑 → 𝑋 ∈ (𝐿‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem1.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
2 | lcfrlem1.s | . . 3 ⊢ 𝑆 = (Scalar‘𝑈) | |
3 | lcfrlem1.q | . . 3 ⊢ × = (.r‘𝑆) | |
4 | lcfrlem1.z | . . 3 ⊢ 0 = (0g‘𝑆) | |
5 | lcfrlem1.i | . . 3 ⊢ 𝐼 = (invr‘𝑆) | |
6 | lcfrlem1.f | . . 3 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lcfrlem1.d | . . 3 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lcfrlem1.t | . . 3 ⊢ · = ( ·𝑠 ‘𝐷) | |
9 | lcfrlem1.m | . . 3 ⊢ − = (-g‘𝐷) | |
10 | lcfrlem1.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ LVec) | |
11 | lcfrlem1.e | . . 3 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
12 | lcfrlem1.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
13 | lcfrlem1.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
14 | lcfrlem1.n | . . 3 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) | |
15 | lcfrlem1.h | . . 3 ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) | |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | lcfrlem1 37698 | . 2 ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
17 | lcfrlem2.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
18 | lveclmod 19501 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
19 | 10, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
20 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
21 | 2 | lmodring 19263 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
22 | 19, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring) |
23 | 2 | lvecdrng 19500 | . . . . . . . . 9 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing) |
24 | 10, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ DivRing) |
25 | 2, 20, 1, 6 | lflcl 35220 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
26 | 10, 12, 13, 25 | syl3anc 1439 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
27 | 20, 4, 5 | drnginvrcl 19156 | . . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
28 | 24, 26, 14, 27 | syl3anc 1439 | . . . . . . 7 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
29 | 2, 20, 1, 6 | lflcl 35220 | . . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
30 | 10, 11, 13, 29 | syl3anc 1439 | . . . . . . 7 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
31 | 20, 3 | ringcl 18948 | . . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
32 | 22, 28, 30, 31 | syl3anc 1439 | . . . . . 6 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
33 | 6, 2, 20, 7, 8, 19, 32, 12 | ldualvscl 35295 | . . . . 5 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹) |
34 | 6, 7, 9, 19, 11, 33 | ldualvsubcl 35312 | . . . 4 ⊢ (𝜑 → (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) ∈ 𝐹) |
35 | 15, 34 | syl5eqel 2863 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
36 | 1, 2, 4, 6, 17, 10, 35, 13 | ellkr2 35247 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐻) ↔ (𝐻‘𝑋) = 0 )) |
37 | 16, 36 | mpbird 249 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐿‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ‘cfv 6135 (class class class)co 6922 Basecbs 16255 .rcmulr 16339 Scalarcsca 16341 ·𝑠 cvsca 16342 0gc0g 16486 -gcsg 17811 Ringcrg 18934 invrcinvr 19058 DivRingcdr 19139 LModclmod 19255 LVecclvec 19497 LFnlclfn 35213 LKerclk 35241 LDualcld 35279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-sbg 17814 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-drng 19141 df-lmod 19257 df-lvec 19498 df-lfl 35214 df-lkr 35242 df-ldual 35280 |
This theorem is referenced by: lcfrlem35 37733 |
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