lcfrlem37 - Mathbox for Norm Megill

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Theorem lcfrlem37 42839

Description: Lemma for lcfr 42845. (Contributed by NM, 8-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem17.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem17.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v	⊢ 𝑉 = (Base‘𝑈)
lcfrlem17.p	⊢ + = (+_g‘𝑈)
lcfrlem17.z	⊢ 0 = (0_g‘𝑈)
lcfrlem17.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem17.a	⊢ 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem17.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem22.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem24.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcfrlem24.s	⊢ 𝑆 = (Scalar‘𝑈)
lcfrlem24.q	⊢ 𝑄 = (0_g‘𝑆)
lcfrlem24.r	⊢ 𝑅 = (Base‘𝑆)
lcfrlem24.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcfrlem24.ib	⊢ (𝜑 → 𝐼 ∈ 𝐵)
lcfrlem24.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem25.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem28.jn	⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
lcfrlem29.i	⊢ 𝐹 = (inv_r‘𝑆)
lcfrlem30.m	⊢ − = (-_g‘𝐷)
lcfrlem30.c	⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
lcfrlem37.g	⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
lcfrlem37.gs	⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
lcfrlem37.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem37.xe	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem37.ye	⊢ (𝜑 → 𝑌 ∈ 𝐸)

**Assertion**
Ref	Expression
lcfrlem37	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

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

**Proof of Theorem lcfrlem37**
Step	Hyp	Ref	Expression
1		lcfrlem30.c	. . . . 5 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
2		lcfrlem25.d	. . . . . 6 ⊢ 𝐷 = (LDual‘𝑈)
3		lcfrlem30.m	. . . . . 6 ⊢ − = (-_g‘𝐷)
4		eqid 2763	. . . . . 6 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
5		lcfrlem17.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
6		lcfrlem17.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		lcfrlem17.k	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	dvhlmod 42370	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
9		lcfrlem37.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (LSubSp‘𝐷))
10		lcfrlem17.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
11		lcfrlem17.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
12		lcfrlem17.p	. . . . . . 7 ⊢ + = (+_g‘𝑈)
13		lcfrlem24.t	. . . . . . 7 ⊢ · = ( ·_𝑠 ‘𝑈)
14		lcfrlem24.s	. . . . . . 7 ⊢ 𝑆 = (Scalar‘𝑈)
15		lcfrlem24.r	. . . . . . 7 ⊢ 𝑅 = (Base‘𝑆)
16		lcfrlem17.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
17		eqid 2763	. . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
19		eqid 2763	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
20		eqid 2763	. . . . . . 7 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
21		lcfrlem24.j	. . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
22		lcfrlem37.gs	. . . . . . 7 ⊢ (𝜑 → 𝐺 ⊆ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)})
23		lcfrlem37.e	. . . . . . 7 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
24		lcfrlem37.xe	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
25		lcfrlem17.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
26		eldifsni 4753	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
27	25, 26	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 )
28		eldifsn 4748	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 ))
29	24, 27, 28	sylanbrc 595	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ (𝐸 ∖ { 0 }))
30	5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 29	lcfrlem16 42818	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)
31		eqid 2763	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
32		lcfrlem17.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
33		lcfrlem17.a	. . . . . . . 8 ⊢ 𝐴 = (LSAtoms‘𝑈)
34		lcfrlem17.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
35		lcfrlem17.ne	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
36		lcfrlem22.b	. . . . . . . 8 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
37		lcfrlem24.q	. . . . . . . 8 ⊢ 𝑄 = (0_g‘𝑆)
38		lcfrlem24.ib	. . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
39		lcfrlem28.jn	. . . . . . . 8 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
40		lcfrlem29.i	. . . . . . . 8 ⊢ 𝐹 = (inv_r‘𝑆)
41	5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40	lcfrlem29 42831	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
42		lcfrlem37.ye	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
43		eldifsni 4753	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
44	34, 43	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 )
45		eldifsn 4748	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
46	42, 44, 45	sylanbrc 595	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ (𝐸 ∖ { 0 }))
47	5, 10, 6, 11, 12, 13, 14, 15, 16, 17, 18, 2, 19, 20, 21, 7, 4, 9, 22, 23, 46	lcfrlem16 42818	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺)
48	14, 15, 2, 31, 4, 8, 9, 41, 47	ldualssvscl 40418	. . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) ∈ 𝐺)
49	2, 3, 4, 8, 9, 30, 48	ldualssvsubcl 40419	. . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺)
50	1, 49	eqeltrid 2868	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐺)
51	5, 10, 6, 11, 12, 16, 32, 33, 7, 25, 34, 35, 36, 13, 14, 37, 15, 21, 38, 18, 2, 39, 40, 3, 1	lcfrlem36 42838	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))
52		2fveq3 6916	. . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶)))
53	52	eleq2d 2850	. . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))))
54	53	rspcev 3580	. . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	50, 51, 54	syl2anc 596	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
56		eliun 4958	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
57	55, 56	sylibr 237	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
58	57, 23	eleqtrrdi 2875	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∃wrex 3090 {crab 3416 ⊆ wss 3900 ∖ cdif 4048 ∩ cin 4092 {csn 4588 {cpr 4590 ∪ ciun 4954 ↦ cmpt 5190 ‘cfv 6561 ℩crio 7402 (class class class)co 7446 Basecbs 17419 +_gcplusg 17460 ._rcmulr 17461 Scalarcsca 17463 ·_𝑠 cvsca 17464 0_gc0g 17642 -_gcsg 19178 inv_rcinvr 20650 LSubSpclss 21239 LSpanclspn 21279 LSAtomsclsa 40234 LFnlclfn 40317 LKerclk 40345 LDualcld 40383 HLchlt 40610 LHypclh 41244 DVecHcdvh 42338 ocHcoch 42607

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-riotaBAD 40213

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-tpos 8274 df-undef 8321 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-map 8880 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-n0 12637 df-z 12724 df-uz 12996 df-fz 13670 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-sca 17476 df-vsca 17477 df-0g 17644 df-mre 17788 df-mrc 17789 df-acs 17791 df-proset 18500 df-poset 18519 df-plt 18534 df-lub 18550 df-glb 18551 df-join 18552 df-meet 18553 df-p0 18629 df-p1 18630 df-lat 18638 df-clat 18705 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-grp 19179 df-minusg 19180 df-sbg 19181 df-subg 19365 df-cntz 19564 df-oppg 19593 df-lsm 19883 df-cmn 20029 df-abl 20030 df-mgp 20394 df-rng 20408 df-ur 20441 df-ring 20494 df-oppr 20600 df-dvdsr 20620 df-unit 20621 df-invr 20651 df-dvr 20664 df-nzr 20796 df-rlreg 20979 df-domn 20980 df-drng 21015 df-lmod 21170 df-lss 21240 df-lsp 21280 df-lvec 21411 df-lsatoms 40236 df-lshyp 40237 df-lcv 40279 df-lfl 40318 df-lkr 40346 df-ldual 40384 df-oposet 40436 df-ol 40438 df-oml 40439 df-covers 40526 df-ats 40527 df-atl 40558 df-cvlat 40582 df-hlat 40611 df-llines 40758 df-lplanes 40759 df-lvols 40760 df-lines 40761 df-psubsp 40763 df-pmap 40764 df-padd 41056 df-lhyp 41248 df-laut 41249 df-ldil 41364 df-ltrn 41365 df-trl 41419 df-tgrp 42003 df-tendo 42015 df-edring 42017 df-dveca 42263 df-disoa 42289 df-dvech 42339 df-dib 42399 df-dic 42433 df-dih 42489 df-doch 42608 df-djh 42655

This theorem is used by: lcfrlem38 42840

W3C validator