lcfrlem38 - Mathbox for Norm Megill

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Theorem lcfrlem38 41574

Description: Lemma for lcfr 41579. Combine lcfrlem27 41563 and lcfrlem37 41573. (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 2729	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
9		lcfrlem38.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 480	. . 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 41539	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4750	. . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	adantr 480	. . 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 41539	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4750	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
26	23, 24, 25	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
27	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
28		lcfrlem38.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 480	. . 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 2729	. . 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 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐼 ∈ 𝐵)
38		simpr 484	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆))
39		lcfrlem38.n	. . . 4 ⊢ (𝜑 → 𝐼 ≠ 0 )
40	39	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐼 ≠ 0 )
41	15, 13	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
42	41	adantr 480	. . 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 3987	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
46	45	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
47	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑋 ∈ 𝐸)
48	22	adantr 480	. . 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 41563	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
50	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51	20	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑋 ∈ (𝑉 ∖ { 0 }))
52	26	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
53	28	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
54	36	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐼 ∈ 𝐵)
55		simpr 484	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆))
56		eqid 2729	. . 3 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
57		eqid 2729	. . 3 ⊢ (-_g‘𝐷) = (-_g‘𝐷)
58		eqid 2729	. . 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 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐺 ∈ (LSubSp‘𝐷))
60	45	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
61	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑋 ∈ 𝐸)
62	22	adantr 480	. . 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 41573	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
64	49, 63	pm2.61dane 3012	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 {crab 3405 ∖ cdif 3911 ∩ cin 3913 ⊆ wss 3914 {csn 4589 {cpr 4591 ∪ ciun 4955 ↦ cmpt 5188 ‘cfv 6511 ℩crio 7343 (class class class)co 7387 Basecbs 17179 +_gcplusg 17220 ._rcmulr 17221 Scalarcsca 17223 ·_𝑠 cvsca 17224 0_gc0g 17402 -_gcsg 18867 inv_rcinvr 20296 LSubSpclss 20837 LSpanclspn 20877 LSAtomsclsa 38967 LFnlclfn 39050 LKerclk 39078 LDualcld 39116 HLchlt 39343 LHypclh 39978 DVecHcdvh 41072 ocHcoch 41341

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-riotaBAD 38946

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-of 7653 df-om 7843 df-1st 7968 df-2nd 7969 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-n0 12443 df-z 12530 df-uz 12794 df-fz 13469 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-mre 17547 df-mrc 17548 df-acs 17550 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18391 df-clat 18458 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-grp 18868 df-minusg 18869 df-sbg 18870 df-subg 19055 df-cntz 19249 df-oppg 19278 df-lsm 19566 df-cmn 19712 df-abl 19713 df-mgp 20050 df-rng 20062 df-ur 20091 df-ring 20144 df-oppr 20246 df-dvdsr 20266 df-unit 20267 df-invr 20297 df-dvr 20310 df-nzr 20422 df-rlreg 20603 df-domn 20604 df-drng 20640 df-lmod 20768 df-lss 20838 df-lsp 20878 df-lvec 21010 df-lsatoms 38969 df-lshyp 38970 df-lcv 39012 df-lfl 39051 df-lkr 39079 df-ldual 39117 df-oposet 39169 df-ol 39171 df-oml 39172 df-covers 39259 df-ats 39260 df-atl 39291 df-cvlat 39315 df-hlat 39344 df-llines 39492 df-lplanes 39493 df-lvols 39494 df-lines 39495 df-psubsp 39497 df-pmap 39498 df-padd 39790 df-lhyp 39982 df-laut 39983 df-ldil 40098 df-ltrn 40099 df-trl 40153 df-tgrp 40737 df-tendo 40749 df-edring 40751 df-dveca 40997 df-disoa 41023 df-dvech 41073 df-dib 41133 df-dic 41167 df-dih 41223 df-doch 41342 df-djh 41389

This theorem is referenced by: lcfrlem39 41575

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