lcfrlem37 - Mathbox for Norm Megill

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Theorem lcfrlem37 41544

Description: Lemma for lcfr 41550. (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 2735	. . . . . 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 41075	. . . . . 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 2735	. . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
19		eqid 2735	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
20		eqid 2735	. . . . . . 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 4766	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 )
28		eldifsn 4762	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 ))
29	24, 27, 28	sylanbrc 583	. . . . . . 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 41523	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)
31		eqid 2735	. . . . . . 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 41536	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
42		lcfrlem37.ye	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
43		eldifsni 4766	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
44	34, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 )
45		eldifsn 4762	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
46	42, 44, 45	sylanbrc 583	. . . . . . . 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 41523	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺)
48	14, 15, 2, 31, 4, 8, 9, 41, 47	ldualssvscl 39122	. . . . . 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 39123	. . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺)
50	1, 49	eqeltrid 2838	. . . 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 41543	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))
52		2fveq3 6880	. . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶)))
53	52	eleq2d 2820	. . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))))
54	53	rspcev 3601	. . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	50, 51, 54	syl2anc 584	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
56		eliun 4971	. . 3 ⊢ ((𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)) ↔ ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
57	55, 56	sylibr 234	. 2 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔)))
58	57, 23	eleqtrrdi 2845	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∃wrex 3060 {crab 3415 ∖ cdif 3923 ∩ cin 3925 ⊆ wss 3926 {csn 4601 {cpr 4603 ∪ ciun 4967 ↦ cmpt 5201 ‘cfv 6530 ℩crio 7359 (class class class)co 7403 Basecbs 17226 +_gcplusg 17269 ._rcmulr 17270 Scalarcsca 17272 ·_𝑠 cvsca 17273 0_gc0g 17451 -_gcsg 18916 inv_rcinvr 20345 LSubSpclss 20886 LSpanclspn 20926 LSAtomsclsa 38938 LFnlclfn 39021 LKerclk 39049 LDualcld 39087 HLchlt 39314 LHypclh 39949 DVecHcdvh 41043 ocHcoch 41312

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-riotaBAD 38917

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-tpos 8223 df-undef 8270 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-n0 12500 df-z 12587 df-uz 12851 df-fz 13523 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-mre 17596 df-mrc 17597 df-acs 17599 df-proset 18304 df-poset 18323 df-plt 18338 df-lub 18354 df-glb 18355 df-join 18356 df-meet 18357 df-p0 18433 df-p1 18434 df-lat 18440 df-clat 18507 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19104 df-cntz 19298 df-oppg 19327 df-lsm 19615 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-oppr 20295 df-dvdsr 20315 df-unit 20316 df-invr 20346 df-dvr 20359 df-nzr 20471 df-rlreg 20652 df-domn 20653 df-drng 20689 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lsatoms 38940 df-lshyp 38941 df-lcv 38983 df-lfl 39022 df-lkr 39050 df-ldual 39088 df-oposet 39140 df-ol 39142 df-oml 39143 df-covers 39230 df-ats 39231 df-atl 39262 df-cvlat 39286 df-hlat 39315 df-llines 39463 df-lplanes 39464 df-lvols 39465 df-lines 39466 df-psubsp 39468 df-pmap 39469 df-padd 39761 df-lhyp 39953 df-laut 39954 df-ldil 40069 df-ltrn 40070 df-trl 40124 df-tgrp 40708 df-tendo 40720 df-edring 40722 df-dveca 40968 df-disoa 40994 df-dvech 41044 df-dib 41104 df-dic 41138 df-dih 41194 df-doch 41313 df-djh 41360

This theorem is referenced by: lcfrlem38 41545

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