lcfrlem15 - Mathbox for Norm Megill

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Theorem lcfrlem15 40297

Description: Lemma for lcfr 40325. (Contributed by NM, 9-Mar-2015.)

**Hypotheses**
Ref	Expression
lcf1o.h	⊢ 𝐻 = (LHyp‘𝐾)
lcf1o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcf1o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcf1o.v	⊢ 𝑉 = (Base‘𝑈)
lcf1o.a	⊢ + = (+_g‘𝑈)
lcf1o.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcf1o.s	⊢ 𝑆 = (Scalar‘𝑈)
lcf1o.r	⊢ 𝑅 = (Base‘𝑆)
lcf1o.z	⊢ 0 = (0_g‘𝑈)
lcf1o.f	⊢ 𝐹 = (LFnl‘𝑈)
lcf1o.l	⊢ 𝐿 = (LKer‘𝑈)
lcf1o.d	⊢ 𝐷 = (LDual‘𝑈)
lcf1o.q	⊢ 𝑄 = (0_g‘𝐷)
lcf1o.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcf1o.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcflo.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem10.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))

**Assertion**
Ref	Expression
lcfrlem15	⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))

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

**Proof of Theorem lcfrlem15**
Step	Hyp	Ref	Expression
1		lcf1o.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcf1o.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcflo.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 39850	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcfrlem10.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
6	5	eldifad 3957	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
7		lcf1o.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2732	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
9	7, 8	lspsnid 20555	. . 3 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}))
10	4, 6, 9	syl2anc 584	. 2 ⊢ (𝜑 → 𝑋 ∈ ((LSpan‘𝑈)‘{𝑋}))
11		lcf1o.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
12		lcf1o.a	. . 3 ⊢ + = (+_g‘𝑈)
13		lcf1o.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
14		lcf1o.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
15		lcf1o.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
16		lcf1o.z	. . 3 ⊢ 0 = (0_g‘𝑈)
17		lcf1o.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
18		lcf1o.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
19		lcf1o.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
20		lcf1o.q	. . 3 ⊢ 𝑄 = (0_g‘𝐷)
21		lcf1o.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22		lcf1o.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
23	1, 11, 2, 7, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 3, 5, 8	lcfrlem14 40296	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = ((LSpan‘𝑈)‘{𝑋}))
24	10, 23	eleqtrrd 2836	1 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 {crab 3432 ∖ cdif 3942 {csn 4623 ↦ cmpt 5225 ‘cfv 6533 ℩crio 7349 (class class class)co 7394 Basecbs 17128 +_gcplusg 17181 Scalarcsca 17184 ·_𝑠 cvsca 17185 0_gc0g 17369 LModclmod 20422 LSpanclspn 20533 LFnlclfn 37796 LKerclk 37824 LDualcld 37862 HLchlt 38089 LHypclh 38724 DVecHcdvh 39818 ocHcoch 40087

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-riotaBAD 37692

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-1st 7959 df-2nd 7960 df-tpos 8195 df-undef 8242 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-er 8688 df-map 8807 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-n0 12457 df-z 12543 df-uz 12807 df-fz 13469 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17371 df-proset 18232 df-poset 18250 df-plt 18267 df-lub 18283 df-glb 18284 df-join 18285 df-meet 18286 df-p0 18362 df-p1 18363 df-lat 18369 df-clat 18436 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-submnd 18650 df-grp 18799 df-minusg 18800 df-sbg 18801 df-subg 18977 df-cntz 19149 df-lsm 19470 df-cmn 19616 df-abl 19617 df-mgp 19949 df-ur 19966 df-ring 20018 df-oppr 20104 df-dvdsr 20125 df-unit 20126 df-invr 20156 df-dvr 20167 df-drng 20269 df-lmod 20424 df-lss 20494 df-lsp 20534 df-lvec 20665 df-lsatoms 37715 df-lshyp 37716 df-lfl 37797 df-lkr 37825 df-oposet 37915 df-ol 37917 df-oml 37918 df-covers 38005 df-ats 38006 df-atl 38037 df-cvlat 38061 df-hlat 38090 df-llines 38238 df-lplanes 38239 df-lvols 38240 df-lines 38241 df-psubsp 38243 df-pmap 38244 df-padd 38536 df-lhyp 38728 df-laut 38729 df-ldil 38844 df-ltrn 38845 df-trl 38899 df-tgrp 39483 df-tendo 39495 df-edring 39497 df-dveca 39743 df-disoa 39769 df-dvech 39819 df-dib 39879 df-dic 39913 df-dih 39969 df-doch 40088 df-djh 40135

This theorem is referenced by: lcfrlem16 40298

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