lcfrlem10 - Mathbox for Norm Megill

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Theorem lcfrlem10 41535

Description: Lemma for lcfr 41568. (Contributed by NM, 23-Feb-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
lcfrlem10	⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)

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

**Proof of Theorem lcfrlem10**
Step	Hyp	Ref	Expression
1		lcf1o.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcf1o.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcf1o.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcf1o.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lcf1o.a	. . 3 ⊢ + = (+_g‘𝑈)
6		lcf1o.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
7		lcf1o.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
8		lcf1o.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
9		lcf1o.z	. . 3 ⊢ 0 = (0_g‘𝑈)
10		lcf1o.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
11		lcf1o.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
12		lcf1o.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
13		lcf1o.q	. . 3 ⊢ 𝑄 = (0_g‘𝐷)
14		lcf1o.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
15		lcf1o.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
16		lcflo.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lcfrlem10.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	lcfrlem8 41532	. 2 ⊢ (𝜑 → (𝐽‘𝑋) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))))
19		eqid 2735	. . 3 ⊢ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
20	1, 2, 3, 4, 9, 5, 6, 10, 7, 8, 19, 16, 17	dochflcl 41458	. 2 ⊢ (𝜑 → (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) ∈ 𝐹)
21	18, 20	eqeltrd 2839	1 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 {crab 3433 ∖ cdif 3960 {csn 4631 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 Scalarcsca 17301 ·_𝑠 cvsca 17302 0_gc0g 17486 LFnlclfn 39039 LKerclk 39067 LDualcld 39105 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 ocHcoch 41330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lfl 39040 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378

This theorem is referenced by: lcfrlem12N 41537 lcfrlem16 41541 lcfrlem25 41550 lcfrlem26 41551 lcfrlem28 41553 lcfrlem29 41554 lcfrlem30 41555 lcfrlem31 41556 lcfrlem33 41558 lcfrlem35 41560

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