lcfrlem1 - Mathbox for Norm Megill

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Theorem lcfrlem1 41047

Description: Lemma for lcfr 41090. 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 6903	. 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 2728	. . . 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 20998	. . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12		lcfrlem1.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
13		eqid 2728	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		lcfrlem1.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐷)
15	4	lvecdrng 20997	. . . . . . . 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 38568	. . . . . . . 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 = (0_g‘𝑆)
23		lcfrlem1.i	. . . . . . . 8 ⊢ 𝐼 = (inv_r‘𝑆)
24	13, 22, 23	drnginvrcl 20653	. . . . . . 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 38568	. . . . . . 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 20763	. . . . . 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 38643	. . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹)
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 38661	. . 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 38642	. . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
34		eqid 2728	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35	13, 22, 28, 34, 23	drnginvrr 20657	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
36	16, 20, 21, 35	syl3anc 1368	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
37	36	oveq1d 7441	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1_r‘𝑆) × (𝐸‘𝑋)))
38	4	lmodring 20758	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
39	11, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
40	13, 28	ringass 20200	. . . . . . 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 20212	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
43	39, 27, 42	syl2anc 582	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
44	37, 41, 43	3eqtr3d 2776	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋))
45	33, 44	eqtrd 2768	. . . 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 20759	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp)
48	11, 47	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp)
49	13, 22, 5	grpsubid 18987	. . . 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 2772	. 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 )
52	2, 51	eqtrid 2780	1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 ._rcmulr 17241 Scalarcsca 17243 ·_𝑠 cvsca 17244 0_gc0g 17428 Grpcgrp 18897 -_gcsg 18899 1_rcur 20128 Ringcrg 20180 inv_rcinvr 20333 DivRingcdr 20631 LModclmod 20750 LVecclvec 20994 LFnlclfn 38561 LDualcld 38627

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-0g 17430 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-drng 20633 df-lmod 20752 df-lvec 20995 df-lfl 38562 df-ldual 38628

This theorem is referenced by: lcfrlem3 41049

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