lcfrlem24 - Mathbox for Norm Megill

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Theorem lcfrlem24 42447

Description: Lemma for lcfr 42466. (Contributed by NM, 24-Feb-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)

**Assertion**
Ref	Expression
lcfrlem24	⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))

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) 𝑄(𝑥, 𝑤, 𝑣, 𝑘) 𝑅(𝑤) 𝑆(𝑥, 𝑤, 𝑣) 𝑈(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 lcfrlem24**
Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem17.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.p	. . 3 ⊢ + = (+_g‘𝑈)
6		lcfrlem17.z	. . 3 ⊢ 0 = (0_g‘𝑈)
7		lcfrlem17.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
8		lcfrlem17.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
9		lcfrlem17.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		lcfrlem17.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11		lcfrlem17.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
12		lcfrlem17.ne	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	lcfrlem18 42441	. 2 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
14		lcfrlem24.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑈)
15		lcfrlem24.s	. . . 4 ⊢ 𝑆 = (Scalar‘𝑈)
16		lcfrlem24.r	. . . 4 ⊢ 𝑅 = (Base‘𝑆)
17		eqid 2762	. . . 4 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . 4 ⊢ 𝐿 = (LKer‘𝑈)
19		eqid 2762	. . . 4 ⊢ (LDual‘𝑈) = (LDual‘𝑈)
20		eqid 2762	. . . 4 ⊢ (0_g‘(LDual‘𝑈)) = (0_g‘(LDual‘𝑈))
21		eqid 2762	. . . 4 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
22		lcfrlem24.j	. . . 4 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
23	1, 2, 3, 4, 5, 14, 15, 16, 6, 17, 18, 19, 20, 21, 22, 9, 10	lcfrlem11 42434	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
24	1, 2, 3, 4, 5, 14, 15, 16, 6, 17, 18, 19, 20, 21, 22, 9, 11	lcfrlem11 42434	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) = ( ⊥ ‘{𝑌}))
25	23, 24	ineq12d 4170	. 2 ⊢ (𝜑 → ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})))
26	13, 25	eqtr4d 2800	1 ⊢ (𝜑 → ( ⊥ ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽‘𝑋)) ∩ (𝐿‘(𝐽‘𝑌))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∃wrex 3088 {crab 3414 ∖ cdif 3899 ∩ cin 3901 {csn 4587 {cpr 4589 ↦ cmpt 5190 ‘cfv 6537 ℩crio 7373 (class class class)co 7417 Basecbs 17307 +_gcplusg 17348 Scalarcsca 17351 ·_𝑠 cvsca 17352 0_gc0g 17530 LSpanclspn 21161 LSAtomsclsa 39855 LFnlclfn 39938 LKerclk 39966 LDualcld 40004 HLchlt 40231 LHypclh 40865 DVecHcdvh 41959 ocHcoch 42228

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-riotaBAD 39834

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-n0 12533 df-z 12620 df-uz 12892 df-fz 13566 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-sca 17364 df-vsca 17365 df-0g 17532 df-proset 18388 df-poset 18407 df-plt 18422 df-lub 18438 df-glb 18439 df-join 18440 df-meet 18441 df-p0 18517 df-p1 18518 df-lat 18526 df-clat 18593 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-subg 19252 df-cntz 19450 df-lsm 19769 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-ring 20380 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-invr 20535 df-dvr 20548 df-drng 20898 df-lmod 21052 df-lss 21122 df-lsp 21162 df-lvec 21293 df-lsatoms 39857 df-lshyp 39858 df-lfl 39939 df-lkr 39967 df-oposet 40057 df-ol 40059 df-oml 40060 df-covers 40147 df-ats 40148 df-atl 40179 df-cvlat 40203 df-hlat 40232 df-llines 40379 df-lplanes 40380 df-lvols 40381 df-lines 40382 df-psubsp 40384 df-pmap 40385 df-padd 40677 df-lhyp 40869 df-laut 40870 df-ldil 40985 df-ltrn 40986 df-trl 41040 df-tgrp 41624 df-tendo 41636 df-edring 41638 df-dveca 41884 df-disoa 41910 df-dvech 41960 df-dib 42020 df-dic 42054 df-dih 42110 df-doch 42229 df-djh 42276

This theorem is used by: lcfrlem25 42448 lcfrlem35 42458

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