lcfrlem38 - Mathbox for Norm Megill

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Theorem lcfrlem38 41836

Description: Lemma for lcfr 41841. Combine lcfrlem27 41825 and lcfrlem37 41835. (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 2736	. . 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 41801	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4742	. . . . 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 41801	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4742	. . . . 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 2736	. . 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 2846	. . . 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 3974	. . . 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 41825	. 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 2736	. . 3 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
57		eqid 2736	. . 3 ⊢ (-_g‘𝐷) = (-_g‘𝐷)
58		eqid 2736	. . 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 41835	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
64	49, 63	pm2.61dane 3019	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 {crab 3399 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 {csn 4580 {cpr 4582 ∪ ciun 4946 ↦ cmpt 5179 ‘cfv 6492 ℩crio 7314 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 ._rcmulr 17178 Scalarcsca 17180 ·_𝑠 cvsca 17181 0_gc0g 17359 -_gcsg 18865 inv_rcinvr 20323 LSubSpclss 20882 LSpanclspn 20922 LSAtomsclsa 39230 LFnlclfn 39313 LKerclk 39341 LDualcld 39379 HLchlt 39606 LHypclh 40240 DVecHcdvh 41334 ocHcoch 41603

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-riotaBAD 39209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-undef 8215 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-n0 12402 df-z 12489 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-0g 17361 df-mre 17505 df-mrc 17506 df-acs 17508 df-proset 18217 df-poset 18236 df-plt 18251 df-lub 18267 df-glb 18268 df-join 18269 df-meet 18270 df-p0 18346 df-p1 18347 df-lat 18355 df-clat 18422 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cntz 19246 df-oppg 19275 df-lsm 19565 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-nzr 20446 df-rlreg 20627 df-domn 20628 df-drng 20664 df-lmod 20813 df-lss 20883 df-lsp 20923 df-lvec 21055 df-lsatoms 39232 df-lshyp 39233 df-lcv 39275 df-lfl 39314 df-lkr 39342 df-ldual 39380 df-oposet 39432 df-ol 39434 df-oml 39435 df-covers 39522 df-ats 39523 df-atl 39554 df-cvlat 39578 df-hlat 39607 df-llines 39754 df-lplanes 39755 df-lvols 39756 df-lines 39757 df-psubsp 39759 df-pmap 39760 df-padd 40052 df-lhyp 40244 df-laut 40245 df-ldil 40360 df-ltrn 40361 df-trl 40415 df-tgrp 40999 df-tendo 41011 df-edring 41013 df-dveca 41259 df-disoa 41285 df-dvech 41335 df-dib 41395 df-dic 41429 df-dih 41485 df-doch 41604 df-djh 41651

This theorem is referenced by: lcfrlem39 41837

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