lcfrlem1 - Mathbox for Norm Megill

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Theorem lcfrlem1 41544

Description: Lemma for lcfr 41587. 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 6907	. 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 2737	. . . 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 21105	. . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12		lcfrlem1.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
13		eqid 2737	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		lcfrlem1.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐷)
15	4	lvecdrng 21104	. . . . . . . 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 39065	. . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆))
20	9, 17, 18, 19	syl3anc 1373	. . . . . . 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 20753	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
25	16, 20, 21, 24	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
26	4, 13, 3, 6	lflcl 39065	. . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆))
27	9, 12, 18, 26	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆))
28		lcfrlem1.q	. . . . . . 7 ⊢ × = (._r‘𝑆)
29	4, 13, 28	lmodmcl 20871	. . . . . 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 1373	. . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆))
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 39140	. . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹)
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 39158	. . 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 39139	. . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
34		eqid 2737	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35	13, 22, 28, 34, 23	drnginvrr 20757	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
36	16, 20, 21, 35	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
37	36	oveq1d 7446	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1_r‘𝑆) × (𝐸‘𝑋)))
38	4	lmodring 20866	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
39	11, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
40	13, 28	ringass 20250	. . . . . . 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 1374	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
42	13, 28, 34	ringlidm 20266	. . . . . . 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 2785	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋))
45	33, 44	eqtrd 2777	. . . 4 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = (𝐸‘𝑋))
46	45	oveq2d 7447	. . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-_g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)))
47	4	lmodfgrp 20867	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp)
48	11, 47	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp)
49	13, 22, 5	grpsubid 19042	. . . 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 2781	. 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 )
52	2, 51	eqtrid 2789	1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ._rcmulr 17298 Scalarcsca 17300 ·_𝑠 cvsca 17301 0_gc0g 17484 Grpcgrp 18951 -_gcsg 18953 1_rcur 20178 Ringcrg 20230 inv_rcinvr 20387 DivRingcdr 20729 LModclmod 20858 LVecclvec 21101 LFnlclfn 39058 LDualcld 39124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-sbg 18956 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 df-drng 20731 df-lmod 20860 df-lvec 21102 df-lfl 39059 df-ldual 39125

This theorem is referenced by: lcfrlem3 41546

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