lcfrlem29 - Mathbox for Norm Megill

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Theorem lcfrlem29 37727

Description: Lemma for lcfr 37741. (Contributed by NM, 9-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem17.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem17.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem17.p	⊢ + = (+_g‘𝑈)
lcfrlem17.z	⊢ 0 = (0_g‘𝑈)
lcfrlem17.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem17.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem17.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem22.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem24.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcfrlem24.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfrlem24.q	⊢ 𝑄 = (0_g‘𝑆)
lcfrlem24.r	⊢ 𝑅 = (Base‘𝑆)
lcfrlem24.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcfrlem24.ib	⊢ (𝜑 → 𝐼 ∈ 𝐵)
lcfrlem24.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem25.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem28.jn	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
lcfrlem29.i	⊢ 𝐹 = (inv_r‘𝑆)

**Assertion**
Ref	Expression
lcfrlem29	⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)

Distinct variable groups: 𝑣,𝑘,𝑤,𝑥, ⊥ + ,𝑘,𝑣,𝑤,𝑥 𝑅,𝑘,𝑣,𝑥 𝑆,𝑘 · ,𝑘,𝑣,𝑤,𝑥 𝑣,𝑉,𝑥 𝑘,𝑋,𝑣,𝑤,𝑥 𝑘,𝑌,𝑣,𝑤,𝑥 𝑥, 0 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑘) 𝐴(𝑥,𝑤,𝑣,𝑘) 𝐵(𝑥,𝑤,𝑣,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑘) 𝑄(𝑥,𝑤,𝑣,𝑘) 𝑅(𝑤) 𝑆(𝑥,𝑤,𝑣) 𝑈(𝑥,𝑤,𝑣,𝑘) 𝐹(𝑥,𝑤,𝑣,𝑘) 𝐻(𝑥,𝑤,𝑣,𝑘) 𝐼(𝑥,𝑤,𝑣,𝑘) 𝐽(𝑥,𝑤,𝑣,𝑘) 𝐾(𝑥,𝑤,𝑣,𝑘) 𝐿(𝑥,𝑤,𝑣,𝑘) 𝑁(𝑥,𝑤,𝑣,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑘) 0 (𝑤,𝑣,𝑘)

**Proof of Theorem lcfrlem29**
Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcfrlem17.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 37266	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcfrlem24.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑈)
6	5	lmodring 19263	. . 3 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
8	1, 2, 3	dvhlvec 37265	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
9	5	lvecdrng 19500	. . . 4 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
10	8, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ DivRing)
11		lcfrlem17.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
12		lcfrlem17.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
13		lcfrlem17.p	. . . . 5 ⊢ + = (+_g‘𝑈)
14		lcfrlem24.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑈)
15		lcfrlem24.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
16		lcfrlem17.z	. . . . 5 ⊢ 0 = (0_g‘𝑈)
17		eqid 2778	. . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
19		lcfrlem25.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
20		eqid 2778	. . . . 5 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
21		eqid 2778	. . . . 5 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22		lcfrlem24.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
23		lcfrlem17.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
24	1, 11, 2, 12, 13, 14, 5, 15, 16, 17, 18, 19, 20, 21, 22, 3, 23	lcfrlem10 37708	. . . 4 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
25		eqid 2778	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
26		lcfrlem17.a	. . . . . 6 ⊢ 𝐴 = (LSAtoms‘𝑈)
27		lcfrlem17.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
28		lcfrlem17.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
29		lcfrlem17.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		lcfrlem22.b	. . . . . . 7 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
31	1, 11, 2, 12, 13, 16, 27, 26, 3, 28, 23, 29, 30	lcfrlem22 37720	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
32	25, 26, 4, 31	lsatlssel 35153	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑈))
33		lcfrlem24.ib	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
34	12, 25	lssel 19330	. . . . 5 ⊢ ((𝐵 ∈ (LSubSp‘𝑈) ∧ 𝐼 ∈ 𝐵) → 𝐼 ∈ 𝑉)
35	32, 33, 34	syl2anc 579	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
36	5, 15, 12, 17	lflcl 35220	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
37	4, 24, 35, 36	syl3anc 1439	. . 3 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
38		lcfrlem28.jn	. . 3 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
39		lcfrlem24.q	. . . 4 ⊢ 𝑄 = (0_g‘𝑆)
40		lcfrlem29.i	. . . 4 ⊢ 𝐹 = (inv_r‘𝑆)
41	15, 39, 40	drnginvrcl 19156	. . 3 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
42	10, 37, 38, 41	syl3anc 1439	. 2 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
43	1, 11, 2, 12, 13, 14, 5, 15, 16, 17, 18, 19, 20, 21, 22, 3, 28	lcfrlem10 37708	. . 3 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
44	5, 15, 12, 17	lflcl 35220	. . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
45	4, 43, 35, 44	syl3anc 1439	. 2 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
46		eqid 2778	. . 3 ⊢ (._r‘𝑆) = (._r‘𝑆)
47	15, 46	ringcl 18948	. 2 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅 ∧ ((𝐽‘𝑋)‘𝐼) ∈ 𝑅) → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
48	7, 42, 45, 47	syl3anc 1439	1 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 ∃wrex 3091 {crab 3094 ∖ cdif 3789 ∩ cin 3791 {csn 4398 {cpr 4400 ↦ cmpt 4965 ‘cfv 6135 ℩crio 6882 (class class class)co 6922 Basecbs 16255 +_gcplusg 16338 ._rcmulr 16339 Scalarcsca 16341 ·_𝑠 cvsca 16342 0_gc0g 16486 Ringcrg 18934 inv_rcinvr 19058 DivRingcdr 19139 LModclmod 19255 LSubSpclss 19324 LSpanclspn 19366 LVecclvec 19497 LSAtomsclsa 35130 LFnlclfn 35213 LKerclk 35241 LDualcld 35279 HLchlt 35506 LHypclh 36140 DVecHcdvh 37234 ocHcoch 37503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-riotaBAD 35109

This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-undef 7681 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-n0 11643 df-z 11729 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-sca 16354 df-vsca 16355 df-0g 16488 df-mre 16632 df-mrc 16633 df-acs 16635 df-proset 17314 df-poset 17332 df-plt 17344 df-lub 17360 df-glb 17361 df-join 17362 df-meet 17363 df-p0 17425 df-p1 17426 df-lat 17432 df-clat 17494 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-grp 17812 df-minusg 17813 df-sbg 17814 df-subg 17975 df-cntz 18133 df-oppg 18159 df-lsm 18435 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-oppr 19010 df-dvdsr 19028 df-unit 19029 df-invr 19059 df-dvr 19070 df-drng 19141 df-lmod 19257 df-lss 19325 df-lsp 19367 df-lvec 19498 df-lsatoms 35132 df-lshyp 35133 df-lcv 35175 df-lfl 35214 df-oposet 35332 df-ol 35334 df-oml 35335 df-covers 35422 df-ats 35423 df-atl 35454 df-cvlat 35478 df-hlat 35507 df-llines 35654 df-lplanes 35655 df-lvols 35656 df-lines 35657 df-psubsp 35659 df-pmap 35660 df-padd 35952 df-lhyp 36144 df-laut 36145 df-ldil 36260 df-ltrn 36261 df-trl 36315 df-tgrp 36899 df-tendo 36911 df-edring 36913 df-dveca 37159 df-disoa 37185 df-dvech 37235 df-dib 37295 df-dic 37329 df-dih 37385 df-doch 37504 df-djh 37551

This theorem is referenced by: lcfrlem30 37728 lcfrlem31 37729 lcfrlem37 37735

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