lcfrlem1 - Mathbox for Norm Megill

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Theorem lcfrlem1 42002

Description: Lemma for lcfr 42045. 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 6835	. 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 21093	. . . . 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 21092	. . . . . . . 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 39524	. . . . . . . 8 ⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆))
20	9, 17, 18, 19	syl3anc 1374	. . . . . . 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 20721	. . . . . . 7 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
25	16, 20, 21, 24	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆))
26	4, 13, 3, 6	lflcl 39524	. . . . . . 7 ⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆))
27	9, 12, 18, 26	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆))
28		lcfrlem1.q	. . . . . . 7 ⊢ × = (._r‘𝑆)
29	4, 13, 28	lmodmcl 20859	. . . . . 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 1374	. . . . 5 ⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆))
31	6, 4, 13, 7, 14, 11, 30, 17	ldualvscl 39599	. . . 4 ⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹)
32	3, 4, 5, 6, 7, 8, 11, 12, 31, 18	ldualvsubval 39617	. . 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 39598	. . . . 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 20725	. . . . . . . 8 ⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
36	16, 20, 21, 35	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1_r‘𝑆))
37	36	oveq1d 7375	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1_r‘𝑆) × (𝐸‘𝑋)))
38	4	lmodring 20854	. . . . . . . 8 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
39	11, 38	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ Ring)
40	13, 28	ringass 20225	. . . . . . 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 1375	. . . . . 6 ⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))))
42	13, 28, 34	ringlidm 20241	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1_r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋))
43	39, 27, 42	syl2anc 585	. . . . . 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 7376	. . 3 ⊢ (𝜑 → ((𝐸‘𝑋)(-_g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)))
47	4	lmodfgrp 20855	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp)
48	11, 47	syl 17	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Grp)
49	13, 22, 5	grpsubid 18991	. . . 4 ⊢ ((𝑆 ∈ Grp ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐸‘𝑋)(-_g‘𝑆)(𝐸‘𝑋)) = 0 )
50	48, 27, 49	syl2anc 585	. . 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 1542 ∈ wcel 2114 ≠ wne 2933 ‘cfv 6492 (class class class)co 7360 Basecbs 17170 ._rcmulr 17212 Scalarcsca 17214 ·_𝑠 cvsca 17215 0_gc0g 17393 Grpcgrp 18900 -_gcsg 18902 1_rcur 20153 Ringcrg 20205 inv_rcinvr 20358 DivRingcdr 20697 LModclmod 20846 LVecclvec 21089 LFnlclfn 39517 LDualcld 39583

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8169 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-er 8636 df-map 8768 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18903 df-minusg 18904 df-sbg 18905 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-drng 20699 df-lmod 20848 df-lvec 21090 df-lfl 39518 df-ldual 39584

This theorem is referenced by: lcfrlem3 42004

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