lcfrlem29 - Mathbox for Norm Megill

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Theorem lcfrlem29 41038

Description: Lemma for lcfr 41052. (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 40577	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcfrlem24.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑈)
6	5	lmodring 20744	. . 3 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
7	4, 6	syl 17	. 2 ⊢ (𝜑 → 𝑆 ∈ Ring)
8	1, 2, 3	dvhlvec 40576	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LVec)
9	5	lvecdrng 20983	. . . 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 2728	. . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
19		lcfrlem25.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
20		eqid 2728	. . . . 5 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
21		eqid 2728	. . . . 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 41019	. . . 4 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
25		eqid 2728	. . . . . 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 41031	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
32	25, 26, 4, 31	lsatlssel 38463	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (LSubSp‘𝑈))
33		lcfrlem24.ib	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
34	12, 25	lssel 20814	. . . . 5 ⊢ ((𝐵 ∈ (LSubSp‘𝑈) ∧ 𝐼 ∈ 𝐵) → 𝐼 ∈ 𝑉)
35	32, 33, 34	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
36	5, 15, 12, 17	lflcl 38530	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
37	4, 24, 35, 36	syl3anc 1369	. . 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 20639	. . 3 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
42	10, 37, 38, 41	syl3anc 1369	. 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 41019	. . 3 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
44	5, 15, 12, 17	lflcl 38530	. . 3 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
45	4, 43, 35, 44	syl3anc 1369	. 2 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) ∈ 𝑅)
46		eqid 2728	. . 3 ⊢ (._r‘𝑆) = (._r‘𝑆)
47	15, 46	ringcl 20183	. 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 1369	1 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∃wrex 3066 {crab 3428 ∖ cdif 3942 ∩ cin 3944 {csn 4624 {cpr 4626 ↦ cmpt 5225 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 Basecbs 17173 +_gcplusg 17226 ._rcmulr 17227 Scalarcsca 17229 ·_𝑠 cvsca 17230 0_gc0g 17414 Ringcrg 20166 inv_rcinvr 20319 DivRingcdr 20617 LModclmod 20736 LSubSpclss 20808 LSpanclspn 20848 LVecclvec 20980 LSAtomsclsa 38440 LFnlclfn 38523 LKerclk 38551 LDualcld 38589 HLchlt 38816 LHypclh 39451 DVecHcdvh 40545 ocHcoch 40814

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38419

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-mre 17559 df-mrc 17560 df-acs 17562 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-oppg 19290 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38442 df-lshyp 38443 df-lcv 38485 df-lfl 38524 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tgrp 40210 df-tendo 40222 df-edring 40224 df-dveca 40470 df-disoa 40496 df-dvech 40546 df-dib 40606 df-dic 40640 df-dih 40696 df-doch 40815 df-djh 40862

This theorem is referenced by: lcfrlem30 41039 lcfrlem31 41040 lcfrlem37 41046

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