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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem1 | Structured version Visualization version GIF version |
Description: Lemma for lcfr 41286. Note that 𝑋 is z in Mario's notes. (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)) |
Ref | Expression |
---|---|
lcfrlem1 | ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcfrlem1.h | . . 3 ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) | |
2 | 1 | fveq1i 6904 | . 2 ⊢ (𝐻‘𝑋) = ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) |
3 | lcfrlem1.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | lcfrlem1.s | . . . 4 ⊢ 𝑆 = (Scalar‘𝑈) | |
5 | eqid 2726 | . . . 4 ⊢ (-g‘𝑆) = (-g‘𝑆) | |
6 | lcfrlem1.f | . . . 4 ⊢ 𝐹 = (LFnl‘𝑈) | |
7 | lcfrlem1.d | . . . 4 ⊢ 𝐷 = (LDual‘𝑈) | |
8 | lcfrlem1.m | . . . 4 ⊢ − = (-g‘𝐷) | |
9 | lcfrlem1.u | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) | |
10 | lveclmod 21086 | . . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | lcfrlem1.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
13 | eqid 2726 | . . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆) | |
14 | lcfrlem1.t | . . . . 5 ⊢ · = ( ·𝑠 ‘𝐷) | |
15 | 4 | lvecdrng 21085 | . . . . . . . 8 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing) |
16 | 9, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ DivRing) |
17 | lcfrlem1.g | . . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
18 | lcfrlem1.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
19 | 4, 13, 3, 6 | lflcl 38764 | . . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
20 | 9, 17, 18, 19 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
21 | lcfrlem1.n | . . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) | |
22 | lcfrlem1.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑆) | |
23 | lcfrlem1.i | . . . . . . . 8 ⊢ 𝐼 = (invr‘𝑆) | |
24 | 13, 22, 23 | drnginvrcl 20733 | . . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
25 | 16, 20, 21, 24 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
26 | 4, 13, 3, 6 | lflcl 38764 | . . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
27 | 9, 12, 18, 26 | syl3anc 1368 | . . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
28 | lcfrlem1.q | . . . . . . 7 ⊢ × = (.r‘𝑆) | |
29 | 4, 13, 28 | lmodmcl 20851 | . . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
30 | 11, 25, 27, 29 | syl3anc 1368 | . . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
31 | 6, 4, 13, 7, 14, 11, 30, 17 | ldualvscl 38839 | . . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹) |
32 | 3, 4, 5, 6, 7, 8, 11, 12, 31, 18 | ldualvsubval 38857 | . . 3 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = ((𝐸‘𝑋)(-g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋))) |
33 | 6, 3, 4, 13, 28, 7, 14, 9, 30, 17, 18 | ldualvsval 38838 | . . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
34 | eqid 2726 | . . . . . . . . 9 ⊢ (1r‘𝑆) = (1r‘𝑆) | |
35 | 13, 22, 28, 34, 23 | drnginvrr 20737 | . . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
36 | 16, 20, 21, 35 | syl3anc 1368 | . . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
37 | 36 | oveq1d 7441 | . . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1r‘𝑆) × (𝐸‘𝑋))) |
38 | 4 | lmodring 20846 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
39 | 11, 38 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring) |
40 | 13, 28 | ringass 20238 | . . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ ((𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆))) → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
41 | 39, 20, 25, 27, 40 | syl13anc 1369 | . . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
42 | 13, 28, 34 | ringlidm 20250 | . . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋)) |
43 | 39, 27, 42 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → ((1r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋)) |
44 | 37, 41, 43 | 3eqtr3d 2774 | . . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋)) |
45 | 33, 44 | eqtrd 2766 | . . . 4 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = (𝐸‘𝑋)) |
46 | 45 | oveq2d 7442 | . . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋))) |
47 | 4 | lmodfgrp 20847 | . . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp) |
48 | 11, 47 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp) |
49 | 13, 22, 5 | grpsubid 19020 | . . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋)) = 0 ) |
50 | 48, 27, 49 | syl2anc 582 | . . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋)) = 0 ) |
51 | 32, 46, 50 | 3eqtrd 2770 | . 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 ) |
52 | 2, 51 | eqtrid 2778 | 1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6556 (class class class)co 7426 Basecbs 17215 .rcmulr 17269 Scalarcsca 17271 ·𝑠 cvsca 17272 0gc0g 17456 Grpcgrp 18930 -gcsg 18932 1rcur 20166 Ringcrg 20218 invrcinvr 20371 DivRingcdr 20709 LModclmod 20838 LVecclvec 21082 LFnlclfn 38757 LDualcld 38823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5292 ax-sep 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-1cn 11218 ax-icn 11219 ax-addcl 11220 ax-addrcl 11221 ax-mulcl 11222 ax-mulrcl 11223 ax-mulcom 11224 ax-addass 11225 ax-mulass 11226 ax-distr 11227 ax-i2m1 11228 ax-1ne0 11229 ax-1rid 11230 ax-rnegex 11231 ax-rrecex 11232 ax-cnre 11233 ax-pre-lttri 11234 ax-pre-lttrn 11235 ax-pre-ltadd 11236 ax-pre-mulgt0 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4916 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-tr 5273 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5639 df-we 5641 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6314 df-ord 6381 df-on 6382 df-lim 6383 df-suc 6384 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-om 7879 df-1st 8005 df-2nd 8006 df-tpos 8243 df-frecs 8298 df-wrecs 8329 df-recs 8403 df-rdg 8442 df-1o 8498 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-fin 8980 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-sub 11498 df-neg 11499 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12613 df-uz 12877 df-fz 13541 df-struct 17151 df-sets 17168 df-slot 17186 df-ndx 17198 df-base 17216 df-ress 17245 df-plusg 17281 df-mulr 17282 df-sca 17284 df-vsca 17285 df-0g 17458 df-mgm 18635 df-sgrp 18714 df-mnd 18730 df-grp 18933 df-minusg 18934 df-sbg 18935 df-cmn 19782 df-abl 19783 df-mgp 20120 df-rng 20138 df-ur 20167 df-ring 20220 df-oppr 20318 df-dvdsr 20341 df-unit 20342 df-invr 20372 df-drng 20711 df-lmod 20840 df-lvec 21083 df-lfl 38758 df-ldual 38824 |
This theorem is referenced by: lcfrlem3 41245 |
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