lcfrlem1 - Mathbox for Norm Megill

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Theorem lcfrlem1 40401

Description: Lemma for lcfr 40444. Note that 𝑋 is z in Mario's notes. (Contributed by NM, 27-Feb-2015.)

**Hypotheses**
Ref	Expression
lcfrlem1.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem1.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfrlem1.q	⊢ × = (._r‘𝑆)
lcfrlem1.z	⊢ 0 = (0_g‘𝑆)
lcfrlem1.i	⊢ 𝐼 = (inv_r‘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))

**Assertion**
Ref	Expression
lcfrlem1	⊢ (𝜑 → (𝐻‘𝑋) = 0 )

**Proof of Theorem lcfrlem1**
Step	Hyp	Ref	Expression
1		lcfrlem1.h	. . 3 ⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))
2	1	fveq1i 6889	. 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 2732	. . . 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 20709	. . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12		lcfrlem1.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
13		eqid 2732	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		lcfrlem1.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐷)
15	4	lvecdrng 20708	. . . . . . . 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 37922	. . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆))
20	9, 17, 18, 19	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑋) ∈ (Base‘𝑆))
21		lcfrlem1.n	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 )
22		lcfrlem1.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑆)
23		lcfrlem1.i	. . . . . . . 8 ⊢ 𝐼 = (inv_r‘𝑆)
24	13, 22, 23	drnginvrcl 20329	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
25	16, 20, 21, 24	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
26	4, 13, 3, 6	lflcl 37922	. . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆))
27	9, 12, 18, 26	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆))
28		lcfrlem1.q	. . . . . . 7 ⊢ × = (._r‘𝑆)
29	4, 13, 28	lmodmcl 20476	. . . . . 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 1371	. . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆))
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 37997	. . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹)
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 38015	. . 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 37996	. . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
34		eqid 2732	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35	13, 22, 28, 34, 23	drnginvrr 20333	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
36	16, 20, 21, 35	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
37	36	oveq1d 7420	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1_r‘𝑆) × (𝐸‘𝑋)))
38	4	lmodring 20471	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
39	11, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
40	13, 28	ringass 20069	. . . . . . 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 1372	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
42	13, 28, 34	ringlidm 20079	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
43	39, 27, 42	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
44	37, 41, 43	3eqtr3d 2780	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋))
45	33, 44	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = (𝐸‘𝑋))
46	45	oveq2d 7421	. . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-_g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)))
47	4	lmodfgrp 20472	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp)
48	11, 47	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp)
49	13, 22, 5	grpsubid 18903	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)) = 0 )
50	48, 27, 49	syl2anc 584	. . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)) = 0 )
51	32, 46, 50	3eqtrd 2776	. 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 )
52	2, 51	eqtrid 2784	1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 ._rcmulr 17194 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 Grpcgrp 18815 -_gcsg 18817 1_rcur 19998 Ringcrg 20049 inv_rcinvr 20193 DivRingcdr 20307 LModclmod 20463 LVecclvec 20705 LFnlclfn 37915 LDualcld 37981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-drng 20309 df-lmod 20465 df-lvec 20706 df-lfl 37916 df-ldual 37982

This theorem is referenced by: lcfrlem3 40403

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