lcfrlem38 - Mathbox for Norm Megill

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Theorem lcfrlem38 41539

Description: Lemma for lcfr 41544. Combine lcfrlem27 41528 and lcfrlem37 41538. (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 2740	. . 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 41504	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4811	. . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 582	. . . 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 41504	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4811	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
26	23, 24, 25	sylanbrc 582	. . . 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 2740	. . 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 2854	. . . 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 4059	. . . 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 41528	. 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 2740	. . 3 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
57		eqid 2740	. . 3 ⊢ (-_g‘𝐷) = (-_g‘𝐷)
58		eqid 2740	. . 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 41538	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
64	49, 63	pm2.61dane 3035	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 {crab 3443 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 {csn 4648 {cpr 4650 ∪ ciun 5015 ↦ cmpt 5249 ‘cfv 6575 ℩crio 7405 (class class class)co 7450 Basecbs 17260 +_gcplusg 17313 ._rcmulr 17314 Scalarcsca 17316 ·_𝑠 cvsca 17317 0_gc0g 17501 -_gcsg 18977 inv_rcinvr 20415 LSubSpclss 20954 LSpanclspn 20994 LSAtomsclsa 38932 LFnlclfn 39015 LKerclk 39043 LDualcld 39081 HLchlt 39308 LHypclh 39943 DVecHcdvh 41037 ocHcoch 41306

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-riotaBAD 38911

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-undef 8316 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-0g 17503 df-mre 17646 df-mrc 17647 df-acs 17649 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cntz 19359 df-oppg 19388 df-lsm 19680 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20362 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20541 df-rlreg 20718 df-domn 20719 df-drng 20755 df-lmod 20884 df-lss 20955 df-lsp 20995 df-lvec 21127 df-lsatoms 38934 df-lshyp 38935 df-lcv 38977 df-lfl 39016 df-lkr 39044 df-ldual 39082 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-llines 39457 df-lplanes 39458 df-lvols 39459 df-lines 39460 df-psubsp 39462 df-pmap 39463 df-padd 39755 df-lhyp 39947 df-laut 39948 df-ldil 40063 df-ltrn 40064 df-trl 40118 df-tgrp 40702 df-tendo 40714 df-edring 40716 df-dveca 40962 df-disoa 40988 df-dvech 41038 df-dib 41098 df-dic 41132 df-dih 41188 df-doch 41307 df-djh 41354

This theorem is referenced by: lcfrlem39 41540

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