lcfrlem37 - Mathbox for Norm Megill

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Theorem lcfrlem37 39593

Description: Lemma for lcfr 39599. (Contributed by NM, 8-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)
lcfrlem30.m	⊢ − = (-_g‘𝐷)
lcfrlem30.c	⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
lcfrlem37.g	⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
lcfrlem37.gs	⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
lcfrlem37.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem37.xe	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem37.ye	⊢ (𝜑 → 𝑌 ∈ 𝐸)

**Assertion**
Ref	Expression
lcfrlem37	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

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 𝑓,𝐽 𝑓,𝐿 ⊥ ,𝑓 + ,𝑓 𝑅,𝑓 · ,𝑓 𝑈,𝑓 𝑓,𝑉 𝑓,𝑋 𝑓,𝑌,𝑘,𝑣,𝑤,𝑥,𝑔 𝐶,𝑔,𝑘 𝐷,𝑔,𝑘 𝑔,𝐺,𝑘 𝑔,𝐼,𝑘 𝑓,𝑔,𝐽,𝑘 𝑔,𝐿,𝑘 ⊥ ,𝑔 + ,𝑔 𝑄,𝑔,𝑘 𝑈,𝑘 𝑔,𝑉 𝑔,𝑋 𝑔,𝑌 𝜑,𝑔,𝑘 𝑣,𝑔,𝑤,𝑥 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑓) 𝐴(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐵(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐶(𝑥,𝑤,𝑣,𝑓) 𝐷(𝑥,𝑤,𝑣,𝑓) 𝑄(𝑥,𝑤,𝑣,𝑓) 𝑅(𝑤,𝑔) 𝑆(𝑥,𝑤,𝑣,𝑓,𝑔) · (𝑔) 𝑈(𝑥,𝑤,𝑣,𝑔) 𝐸(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐹(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐺(𝑥,𝑤,𝑣,𝑓) 𝐻(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐼(𝑥,𝑤,𝑣,𝑓) 𝐽(𝑥,𝑤,𝑣) 𝐾(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐿(𝑥,𝑤,𝑣) − (𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑁(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 0 (𝑤,𝑣,𝑓,𝑔,𝑘)

**Proof of Theorem lcfrlem37**
Step	Hyp	Ref	Expression
1		lcfrlem30.c	. . . . 5 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
2		lcfrlem25.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
3		lcfrlem30.m	. . . . . 6 ⊢ − = (-_g‘𝐷)
4		eqid 2738	. . . . . 6 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
5		lcfrlem17.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
6		lcfrlem17.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		lcfrlem17.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	dvhlmod 39124	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
9		lcfrlem37.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
10		lcfrlem17.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11		lcfrlem17.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
12		lcfrlem17.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
13		lcfrlem24.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑈)
14		lcfrlem24.s	. . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑈)
15		lcfrlem24.r	. . . . . . 7 ⊢ 𝑅 = (Base‘𝑆)
16		lcfrlem17.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
17		eqid 2738	. . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
19		eqid 2738	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
20		eqid 2738	. . . . . . 7 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
21		lcfrlem24.j	. . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
22		lcfrlem37.gs	. . . . . . 7 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
23		lcfrlem37.e	. . . . . . 7 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
24		lcfrlem37.xe	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
25		lcfrlem17.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
26		eldifsni 4723	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 )
28		eldifsn 4720	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 ))
29	24, 27, 28	sylanbrc 583	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 }))
30	5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 29	lcfrlem16 39572	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)
31		eqid 2738	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
32		lcfrlem17.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
33		lcfrlem17.a	. . . . . . . 8 ⊢ 𝐴 = (LSAtoms‘𝑈)
34		lcfrlem17.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
35		lcfrlem17.ne	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
36		lcfrlem22.b	. . . . . . . 8 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
37		lcfrlem24.q	. . . . . . . 8 ⊢ 𝑄 = (0_g‘𝑆)
38		lcfrlem24.ib	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
39		lcfrlem28.jn	. . . . . . . 8 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
40		lcfrlem29.i	. . . . . . . 8 ⊢ 𝐹 = (inv_r‘𝑆)
41	5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40	lcfrlem29 39585	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
42		lcfrlem37.ye	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
43		eldifsni 4723	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
44	34, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 )
45		eldifsn 4720	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
46	42, 44, 45	sylanbrc 583	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐸 ∖ { 0 }))
47	5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 46	lcfrlem16 39572	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺)
48	14, 15, 2, 31, 4, 8, 9, 41, 47	ldualssvscl 37172	. . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ 𝐺)
49	2, 3, 4, 8, 9, 30, 48	ldualssvsubcl 37173	. . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺)
50	1, 49	eqeltrid 2843	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐺)
51	5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40, 3, 1	lcfrlem36 39592	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))
52		2fveq3 6779	. . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶)))
53	52	eleq2d 2824	. . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))))
54	53	rspcev 3561	. . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	50, 51, 54	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
56		eliun 4928	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
57	55, 56	sylibr 233	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
58	57, 23	eleqtrrdi 2850	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 {crab 3068 ∖ cdif 3884 ∩ cin 3886 ⊆ wss 3887 {csn 4561 {cpr 4563 ∪ ciun 4924 ↦ cmpt 5157 ‘cfv 6433 ℩crio 7231 (class class class)co 7275 Basecbs 16912 +_gcplusg 16962 ._rcmulr 16963 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 -_gcsg 18579 inv_rcinvr 19913 LSubSpclss 20193 LSpanclspn 20233 LSAtomsclsa 36988 LFnlclfn 37071 LKerclk 37099 LDualcld 37137 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 ocHcoch 39361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409

This theorem is referenced by: lcfrlem38 39594

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