lcfrlem26 - Mathbox for Norm Megill

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Theorem lcfrlem26 42593

Description: Lemma for lcfr 42610. Special case of lcfrlem36 42603 when ((𝐽‘𝑌)‘𝐼) is zero. (Contributed by NM, 11-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)
lcfrlem25.jz	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
lcfrlem25.in	⊢ (𝜑 → 𝐼 ≠ 0 )

**Assertion**
Ref	Expression
lcfrlem26	⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(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) 𝐷(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 lcfrlem26 Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem17.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem17.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcfrlem17.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
4		lcfrlem17.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem17.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
6		lcfrlem17.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		lcfrlem17.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
8		lcfrlem17.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
9		lcfrlem17.a	. . . . . . 7 ⊢ 𝐴 = (LSAtoms‘𝑈)
10		lcfrlem17.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11		lcfrlem17.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
12		lcfrlem17.ne	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
13	1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12	lcfrlem17 42584	. . . . . 6 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3911	. . . . 5 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑉)
15	1, 2, 3, 4, 5, 6, 14	dochocsn 42406	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = (𝑁‘{(𝑋 + 𝑌)}))
16		lcfrlem22.b	. . . . . 6 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
17		lcfrlem24.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝑈)
18		lcfrlem24.s	. . . . . 6 ⊢ 𝑆 = (Scalar‘𝑈)
19		lcfrlem24.q	. . . . . 6 ⊢ 𝑄 = (0_g‘𝑆)
20		lcfrlem24.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝑆)
21		lcfrlem24.j	. . . . . 6 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
22		lcfrlem24.ib	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
23		lcfrlem24.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
24		lcfrlem25.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
25		lcfrlem25.jz	. . . . . 6 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) = 𝑄)
26		lcfrlem25.in	. . . . . 6 ⊢ (𝜑 → 𝐼 ≠ 0 )
27	1, 3, 2, 4, 7, 8, 5, 9, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26	lcfrlem25 42592	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{(𝑋 + 𝑌)}) = (𝐿‘(𝐽‘𝑌)))
28	27	fveq2d 6881	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{(𝑋 + 𝑌)})) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
29	15, 28	eqtr3d 2798	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
30		eqimss 3989	. . 3 ⊢ ((𝑁‘{(𝑋 + 𝑌)}) = ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
31	29, 30	syl 18	. 2 ⊢ (𝜑 → (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))
32		eqid 2761	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
33	1, 2, 6	dvhlmod 42135	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
34		eqid 2761	. . . . 5 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
35		eqid 2761	. . . . . 6 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
36		eqid 2761	. . . . . 6 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
37	1, 3, 2, 4, 7, 17, 18, 20, 8, 34, 23, 24, 35, 36, 21, 6, 11	lcfrlem10 42577	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
38	4, 34, 23, 33, 37	lkrssv 40121	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉)
39	1, 2, 4, 32, 3	dochlss 42379	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐽‘𝑌)) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈))
40	6, 38, 39	syl2anc 596	. . 3 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ∈ (LSubSp‘𝑈))
41	4, 32, 5, 33, 40, 14	ellspsn5b 21250	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))) ↔ (𝑁‘{(𝑋 + 𝑌)}) ⊆ ( ⊥ ‘(𝐿‘(𝐽‘𝑌)))))
42	31, 41	mpbird 260	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘(𝐽‘𝑌))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 {crab 3413 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 {csn 4584 {cpr 4586 ↦ cmpt 5186 ‘cfv 6531 ℩crio 7368 (class class class)co 7412 Basecbs 17367 +_gcplusg 17408 Scalarcsca 17411 ·_𝑠 cvsca 17412 0_gc0g 17590 LSubSpclss 21186 LSpanclspn 21226 LSAtomsclsa 39999 LFnlclfn 40082 LKerclk 40110 LDualcld 40148 HLchlt 40375 LHypclh 41009 DVecHcdvh 42103 ocHcoch 42372

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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-riotaBAD 39978

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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-undef 8274 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-0g 17592 df-mre 17736 df-mrc 17737 df-acs 17739 df-proset 18448 df-poset 18467 df-plt 18482 df-lub 18498 df-glb 18499 df-join 18500 df-meet 18501 df-p0 18577 df-p1 18578 df-lat 18586 df-clat 18653 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-subg 19313 df-cntz 19511 df-oppg 19540 df-lsm 19830 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-drng 20962 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lvec 21358 df-lsatoms 40001 df-lshyp 40002 df-lcv 40044 df-lfl 40083 df-lkr 40111 df-ldual 40149 df-oposet 40201 df-ol 40203 df-oml 40204 df-covers 40291 df-ats 40292 df-atl 40323 df-cvlat 40347 df-hlat 40376 df-llines 40523 df-lplanes 40524 df-lvols 40525 df-lines 40526 df-psubsp 40528 df-pmap 40529 df-padd 40821 df-lhyp 41013 df-laut 41014 df-ldil 41129 df-ltrn 41130 df-trl 41184 df-tgrp 41768 df-tendo 41780 df-edring 41782 df-dveca 42028 df-disoa 42054 df-dvech 42104 df-dib 42164 df-dic 42198 df-dih 42254 df-doch 42373 df-djh 42420

This theorem is used by: lcfrlem27 42594

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