lcfrlem38 - Mathbox for Norm Megill

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Theorem lcfrlem38 38718

Description: Lemma for lcfr 38723. Combine lcfrlem27 38707 and lcfrlem37 38717. (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 2823	. . 3 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
9		lcfrlem38.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	9	adantr 483	. . 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 38683	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
18		lcfrlem38.x	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 0 )
19		eldifsn 4721	. . . . 5 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ))
20	17, 18, 19	sylanbrc 585	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
21	20	adantr 483	. . 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 38683	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
24		lcfrlem38.y	. . . . 5 ⊢ (𝜑 → 𝑌 ≠ 0 )
25		eldifsn 4721	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
26	23, 24, 25	sylanbrc 585	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
27	26	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
28		lcfrlem38.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 483	. . 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 2823	. . 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 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐼 ∈ 𝐵)
38		simpr 487	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆))
39		lcfrlem38.n	. . . 4 ⊢ (𝜑 → 𝐼 ≠ 0 )
40	39	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐼 ≠ 0 )
41	15, 13	eleqtrdi 2925	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
42	41	adantr 483	. . 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 4019	. . . 4 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
46	45	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
47	16	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → 𝑋 ∈ 𝐸)
48	22	adantr 483	. . 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 38707	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) = (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
50	9	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
51	20	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑋 ∈ (𝑉 ∖ { 0 }))
52	26	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑌 ∈ (𝑉 ∖ { 0 }))
53	28	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
54	36	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐼 ∈ 𝐵)
55		simpr 487	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆))
56		eqid 2823	. . 3 ⊢ (inv_r‘𝑆) = (inv_r‘𝑆)
57		eqid 2823	. . 3 ⊢ (-_g‘𝐷) = (-_g‘𝐷)
58		eqid 2823	. . 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 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐺 ∈ (LSubSp‘𝐷))
60	45	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
61	16	adantr 483	. . 3 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → 𝑋 ∈ 𝐸)
62	22	adantr 483	. . 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 38717	. 2 ⊢ ((𝜑 ∧ ((𝐽‘𝑌)‘𝐼) ≠ (0_g‘𝑆)) → (𝑋 + 𝑌) ∈ 𝐸)
64	49, 63	pm2.61dane 3106	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 {crab 3144 ∖ cdif 3935 ∩ cin 3937 ⊆ wss 3938 {csn 4569 {cpr 4571 ∪ ciun 4921 ↦ cmpt 5148 ‘cfv 6357 ℩crio 7115 (class class class)co 7158 Basecbs 16485 +_gcplusg 16567 ._rcmulr 16568 Scalarcsca 16570 ·_𝑠 cvsca 16571 0_gc0g 16715 -_gcsg 18107 inv_rcinvr 19423 LSubSpclss 19705 LSpanclspn 19745 LSAtomsclsa 36112 LFnlclfn 36195 LKerclk 36223 LDualcld 36261 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 ocHcoch 38485

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533

This theorem is referenced by: lcfrlem39 38719

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