lcfrlem38 - Mathbox for Norm Megill

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Theorem lcfrlem38 42216

Description: Lemma for lcfr 42221. Combine lcfrlem27 42205 and lcfrlem37 42215. (Contributed by NM, 11-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem38.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem38.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem38.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem38.p	⊢ + = (+_g‘𝑈)
lcfrlem38.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfrlem38.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem38.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem38.q	⊢ 𝑄 = (LSubSp‘𝐷)
lcfrlem38.c	⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem38.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem38.g	⊢ (𝜑 → 𝐺 ∈ 𝑄)
lcfrlem38.gs	⊢ (𝜑 → 𝐺 ⊆ 𝐶)
lcfrlem38.xe	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem38.ye	⊢ (𝜑 → 𝑌 ∈ 𝐸)
lcfrlem38.z	⊢ 0 = (0_g‘𝑈)
lcfrlem38.x	⊢ (𝜑 → 𝑋 ≠ 0 )
lcfrlem38.y	⊢ (𝜑 → 𝑌 ≠ 0 )
lcfrlem38.sp	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem38.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem38.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem38.i	⊢ (𝜑 → 𝐼 ∈ 𝐵)
lcfrlem38.n	⊢ (𝜑 → 𝐼 ≠ 0 )
lcfrlem38.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem38.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcfrlem38.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfrlem38.r	⊢ 𝑅 = (Base‘𝑆)
lcfrlem38.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))

**Assertion**
Ref	Expression
lcfrlem38	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Distinct variable groups: 𝑔,𝑘,𝐷 𝑔,𝐺,𝑘 𝑔,𝐼,𝑘 𝑓,𝑔,𝑘,𝐽 𝑓,𝐿,𝑔,𝑘 𝑣,𝑓,𝑤,𝑥, ⊥ ,𝑔,𝑘 + ,𝑓,𝑔,𝑘,𝑣,𝑤,𝑥 𝑅,𝑓,𝑘,𝑣,𝑥 𝑆,𝑔,𝑘 · ,𝑓,𝑘,𝑣,𝑤,𝑥 𝑈,𝑓,𝑔,𝑘,𝑣,𝑤,𝑥 𝑓,𝑉,𝑔,𝑣,𝑥 𝑓,𝑋,𝑔,𝑘,𝑣,𝑤,𝑥 𝑓,𝑌,𝑔,𝑘,𝑣,𝑤,𝑥 0 ,𝑓,𝑔,𝑘,𝑥 𝜑,𝑔,𝑘 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑓) 𝐵(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐶(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑓) 𝑄(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑅(𝑤,𝑔) 𝑆(𝑥,𝑤,𝑣,𝑓) · (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) 𝑁(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑓,𝑔,𝑘) 0 (𝑤,𝑣)

**Proof of Theorem lcfrlem38**
Step	Hyp	Ref	Expression
1		lcfrlem38.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem38.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem38.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem38.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		lcfrlem38.p	. . 3 ⊢ + = (+_g‘𝑈)
6		lcfrlem38.z	. . 3 ⊢ 0 = (0_g‘𝑈)
7		lcfrlem38.sp	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
8		eqid 2765	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
9		lcfrlem38.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		lcfrlem38.l	. . . . . 6 ⊢ 𝐿 = (LKer‘𝑈)
12		lcfrlem38.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
13		lcfrlem38.q	. . . . . 6 ⊢ 𝑄 = (LSubSp‘𝐷)
14		lcfrlem38.e	. . . . . 6 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
15		lcfrlem38.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
16		lcfrlem38.xe	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
17	1, 2, 3, 4, 11, 12, 13, 14, 9, 15, 16	lcfrlem4 42181	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4749	. . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 594	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑋 ∈ (𝑉 ∖ { 0 }))
22		lcfrlem38.ye	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
23	1, 2, 3, 4, 11, 12, 13, 14, 9, 15, 22	lcfrlem4 42181	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4749	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
26	23, 24, 25	sylanbrc 594	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
27	26	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
28		lcfrlem38.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		lcfrlem38.b	. . 3 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
31		lcfrlem38.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
32		lcfrlem38.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
33		eqid 2765	. . 3 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
34		lcfrlem38.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
35		lcfrlem38.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
36		lcfrlem38.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
37	36	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐼 ∈ 𝐵)
38		simpr 489	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆))
39		lcfrlem38.n	. . . 4 ⊢ (𝜑 → 𝐼 ≠ 0 )
40	39	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐼 ≠ 0 )
41	15, 13	eleqtrdi 2875	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
42	41	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐺 ∈ (LSubSp‘𝐷))
43		lcfrlem38.gs	. . . . 5 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
44		lcfrlem38.c	. . . . 5 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
45	43, 44	sseqtrdi 3979	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
46	45	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
47	16	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑋 ∈ 𝐸)
48	22	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑌 ∈ 𝐸)
49	1, 2, 3, 4, 5, 6, 7, 8, 10, 21, 27, 29, 30, 31, 32, 33, 34, 35, 37, 11, 12, 38, 40, 42, 46, 14, 47, 48	lcfrlem27 42205	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
50	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51	20	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑋 ∈ (𝑉 ∖ { 0 }))
52	26	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
53	28	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
54	36	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐼 ∈ 𝐵)
55		simpr 489	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆))
56		eqid 2765	. . 3 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
57		eqid 2765	. . 3 ⊢ (-_g‘𝐷) = (-_g‘𝐷)
58		eqid 2765	. . 3 ⊢ ((𝐽‘𝑋)(-_g‘𝐷)((((inv_r‘𝑆)‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) = ((𝐽‘𝑋)(-_g‘𝐷)((((inv_r‘𝑆)‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
59	41	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐺 ∈ (LSubSp‘𝐷))
60	45	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
61	16	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑋 ∈ 𝐸)
62	22	adantr 485	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑌 ∈ 𝐸)
63	1, 2, 3, 4, 5, 6, 7, 8, 50, 51, 52, 53, 30, 31, 32, 33, 34, 35, 54, 11, 12, 55, 56, 57, 58, 59, 60, 14, 61, 62	lcfrlem37 42215	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
64	49, 63	pm2.61dane 3047	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ≠ wne 2960 ∃wrex 3089 {crab 3417 ∖ cdif 3904 ∩ cin 3906 ⊆ wss 3907 {csn 4585 {cpr 4587 ∪ ciun 4952 ↦ cmpt 5186 ‘cfv 6525 ℩crio 7356 (class class class)co 7400 Basecbs 17259 +_gcplusg 17300 ._rcmulr 17301 Scalarcsca 17303 ·_𝑠 cvsca 17304 0_gc0g 17482 -_gcsg 18992 inv_rcinvr 20460 LSubSpclss 21021 LSpanclspn 21061 LSAtomsclsa 39610 LFnlclfn 39693 LKerclk 39721 LDualcld 39759 HLchlt 39986 LHypclh 40620 DVecHcdvh 41714 ocHcoch 41983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-riotaBAD 39589

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-tpos 8210 df-undef 8257 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-n0 12496 df-z 12583 df-uz 12854 df-fz 13527 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-sca 17316 df-vsca 17317 df-0g 17484 df-mre 17628 df-mrc 17629 df-acs 17631 df-proset 18340 df-poset 18359 df-plt 18374 df-lub 18390 df-glb 18391 df-join 18392 df-meet 18393 df-p0 18469 df-p1 18470 df-lat 18478 df-clat 18545 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-grp 18993 df-minusg 18994 df-sbg 18995 df-subg 19180 df-cntz 19378 df-oppg 19407 df-lsm 19697 df-cmn 19843 df-abl 19844 df-mgp 20208 df-rng 20222 df-ur 20255 df-ring 20308 df-oppr 20410 df-dvdsr 20430 df-unit 20431 df-invr 20461 df-dvr 20474 df-nzr 20587 df-rlreg 20770 df-domn 20771 df-drng 20806 df-lmod 20952 df-lss 21022 df-lsp 21062 df-lvec 21193 df-lsatoms 39612 df-lshyp 39613 df-lcv 39655 df-lfl 39694 df-lkr 39722 df-ldual 39760 df-oposet 39812 df-ol 39814 df-oml 39815 df-covers 39902 df-ats 39903 df-atl 39934 df-cvlat 39958 df-hlat 39987 df-llines 40134 df-lplanes 40135 df-lvols 40136 df-lines 40137 df-psubsp 40139 df-pmap 40140 df-padd 40432 df-lhyp 40624 df-laut 40625 df-ldil 40740 df-ltrn 40741 df-trl 40795 df-tgrp 41379 df-tendo 41391 df-edring 41393 df-dveca 41639 df-disoa 41665 df-dvech 41715 df-dib 41775 df-dic 41809 df-dih 41865 df-doch 41984 df-djh 42031

This theorem is referenced by: lcfrlem39 42217

W3C validator