lcfrlem1 - Mathbox for Norm Megill

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Theorem lcfrlem1 42130

Description: Lemma for lcfr 42173. 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 6864	. 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 2761	. . . 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 21153	. . . . 5 ⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
12		lcfrlem1.e	. . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝐹)
13		eqid 2761	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
14		lcfrlem1.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐷)
15	4	lvecdrng 21152	. . . . . . . 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 39652	. . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆))
20	9, 17, 18, 19	syl3anc 1389	. . . . . . 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 20782	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
25	16, 20, 21, 24	syl3anc 1389	. . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
26	4, 13, 3, 6	lflcl 39652	. . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆))
27	9, 12, 18, 26	syl3anc 1389	. . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆))
28		lcfrlem1.q	. . . . . . 7 ⊢ × = (._r‘𝑆)
29	4, 13, 28	lmodmcl 20920	. . . . . 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 1389	. . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆))
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 39727	. . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹)
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 39745	. . 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 39726	. . . . 5 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
34		eqid 2761	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
35	13, 22, 28, 34, 23	drnginvrr 20786	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
36	16, 20, 21, 35	syl3anc 1389	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
37	36	oveq1d 7407	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1_r‘𝑆) × (𝐸‘𝑋)))
38	4	lmodring 20915	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
39	11, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
40	13, 28	ringass 20282	. . . . . . 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 1390	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
42	13, 28, 34	ringlidm 20298	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
43	39, 27, 42	syl2anc 593	. . . . . 6 ⊢ (𝜑 → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
44	37, 41, 43	3eqtr3d 2804	. . . . 5 ⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋))
45	33, 44	eqtrd 2796	. . . 4 ⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = (𝐸‘𝑋))
46	45	oveq2d 7408	. . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-_g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)))
47	4	lmodfgrp 20916	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp)
48	11, 47	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp)
49	13, 22, 5	grpsubid 19049	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)) = 0 )
50	48, 27, 49	syl2anc 593	. . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)) = 0 )
51	32, 46, 50	3eqtrd 2800	. 2 ⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 )
52	2, 51	eqtrid 2808	1 ⊢ (𝜑 → (𝐻‘𝑋) = 0 )

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 ._rcmulr 17270 Scalarcsca 17272 ·_𝑠 cvsca 17273 0_gc0g 17451 Grpcgrp 18958 -_gcsg 18960 1_rcur 20210 Ringcrg 20262 inv_rcinvr 20415 DivRingcdr 20758 LModclmod 20907 LVecclvec 21149 LFnlclfn 39645 LDualcld 39711

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-grp 18961 df-minusg 18962 df-sbg 18963 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-drng 20760 df-lmod 20909 df-lvec 21150 df-lfl 39646 df-ldual 39712

This theorem is referenced by: lcfrlem3 42132

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