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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem3 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 38601. (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 38558 | . 2 ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
17 | lcfrlem2.l | . . 3 ⊢ 𝐿 = (LKer‘𝑈) | |
18 | lveclmod 19807 | . . . . . 6 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
19 | 10, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
20 | eqid 2818 | . . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
21 | 2 | lmodring 19571 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
22 | 19, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring) |
23 | 2 | lvecdrng 19806 | . . . . . . . . 9 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing) |
24 | 10, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ DivRing) |
25 | 2, 20, 1, 6 | lflcl 36080 | . . . . . . . . 9 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
26 | 10, 12, 13, 25 | syl3anc 1363 | . . . . . . . 8 ⊢ (𝜑 → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
27 | 20, 4, 5 | drnginvrcl 19448 | . . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
28 | 24, 26, 14, 27 | syl3anc 1363 | . . . . . . 7 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
29 | 2, 20, 1, 6 | lflcl 36080 | . . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
30 | 10, 11, 13, 29 | syl3anc 1363 | . . . . . . 7 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
31 | 20, 3 | ringcl 19240 | . . . . . . 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 1363 | . . . . . 6 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
33 | 6, 2, 20, 7, 8, 19, 32, 12 | ldualvscl 36155 | . . . . 5 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹) |
34 | 6, 7, 9, 19, 11, 33 | ldualvsubcl 36172 | . . . 4 ⊢ (𝜑 → (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) ∈ 𝐹) |
35 | 15, 34 | eqeltrid 2914 | . . 3 ⊢ (𝜑 → 𝐻 ∈ 𝐹) |
36 | 1, 2, 4, 6, 17, 10, 35, 13 | ellkr2 36107 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐻) ↔ (𝐻‘𝑋) = 0 )) |
37 | 16, 36 | mpbird 258 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐿‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 -gcsg 18043 Ringcrg 19226 invrcinvr 19350 DivRingcdr 19431 LModclmod 19563 LVecclvec 19803 LFnlclfn 36073 LKerclk 36101 LDualcld 36139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-drng 19433 df-lmod 19565 df-lvec 19804 df-lfl 36074 df-lkr 36102 df-ldual 36140 |
This theorem is referenced by: lcfrlem35 38593 |
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