lcfrlem37 - Mathbox for Norm Megill

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Theorem lcfrlem37 38717

Description: Lemma for lcfr 38723. (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 2823	. . . . . 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 38248	. . . . . 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 2823	. . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
19		eqid 2823	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
20		eqid 2823	. . . . . . 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 4724	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 )
28		eldifsn 4721	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 ))
29	24, 27, 28	sylanbrc 585	. . . . . . 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 38696	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)
31		eqid 2823	. . . . . . 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 38709	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
42		lcfrlem37.ye	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
43		eldifsni 4724	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
44	34, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 )
45		eldifsn 4721	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
46	42, 44, 45	sylanbrc 585	. . . . . . . 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 38696	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺)
48	14, 15, 2, 31, 4, 8, 9, 41, 47	ldualssvscl 36296	. . . . . 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 36297	. . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺)
50	1, 49	eqeltrid 2919	. . . 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 38716	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))
52		2fveq3 6677	. . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶)))
53	52	eleq2d 2900	. . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))))
54	53	rspcev 3625	. . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	50, 51, 54	syl2anc 586	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
56		eliun 4925	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
57	55, 56	sylibr 236	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
58	57, 23	eleqtrrdi 2926	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: lcfrlem38 38718

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