lcfrlem37 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfrlem37		Structured version Visualization version GIF version

Theorem lcfrlem37 42167

Description: Lemma for lcfr 42173. (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 2761	. . . . . 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 41698	. . . . . 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 2761	. . . . . . 7 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
18		lcfrlem24.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
19		eqid 2761	. . . . . . 7 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
20		eqid 2761	. . . . . . 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 4749	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ { 0 }) → 𝑋 ≠ 0 )
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 0 )
28		eldifsn 4745	. . . . . . . 8 ⊢ (𝑋 ∈ (𝐸 ∖ { 0 }) ↔ (𝑋 ∈ 𝐸 ∧ 𝑋 ≠ 0 ))
29	24, 27, 28	sylanbrc 592	. . . . . . 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 42146	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐺)
31		eqid 2761	. . . . . . 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 42159	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) ∈ 𝑅)
42		lcfrlem37.ye	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
43		eldifsni 4749	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) → 𝑌 ≠ 0 )
44	34, 43	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ≠ 0 )
45		eldifsn 4745	. . . . . . . . 9 ⊢ (𝑌 ∈ (𝐸 ∖ { 0 }) ↔ (𝑌 ∈ 𝐸 ∧ 𝑌 ≠ 0 ))
46	42, 44, 45	sylanbrc 592	. . . . . . . 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 42146	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ 𝐺)
48	14, 15, 2, 31, 4, 8, 9, 41, 47	ldualssvscl 39746	. . . . . 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 39747	. . . . 5 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) ∈ 𝐺)
50	1, 49	eqeltrid 2865	. . . 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 42166	. . . 4 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶)))
52		2fveq3 6868	. . . . . 6 ⊢ (𝑔 = 𝐶 → ( ⊥ ‘(𝐿‘𝑔)) = ( ⊥ ‘(𝐿‘𝐶)))
53	52	eleq2d 2847	. . . . 5 ⊢ (𝑔 = 𝐶 → ((𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)) ↔ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))))
54	53	rspcev 3581	. . . 4 ⊢ ((𝐶 ∈ 𝐺 ∧ (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝐶))) → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
55	50, 51, 54	syl2anc 593	. . 3 ⊢ (𝜑 → ∃𝑔 ∈ 𝐺 (𝑋 + 𝑌) ∈ ( ⊥ ‘(𝐿‘𝑔)))
56		eliun 4952	. . 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 2872	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3085 {crab 3413 ∖ cdif 3901 ∩ cin 3903 ⊆ wss 3904 {csn 4581 {cpr 4583 ∪ ciun 4948 ↦ cmpt 5180 ‘cfv 6517 ℩crio 7348 (class class class)co 7392 Basecbs 17228 +_gcplusg 17269 ._rcmulr 17270 Scalarcsca 17272 ·_𝑠 cvsca 17273 0_gc0g 17451 -_gcsg 18960 inv_rcinvr 20415 LSubSpclss 20978 LSpanclspn 21018 LSAtomsclsa 39562 LFnlclfn 39645 LKerclk 39673 LDualcld 39711 HLchlt 39938 LHypclh 40572 DVecHcdvh 41666 ocHcoch 41935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-riotaBAD 39541

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-undef 8248 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-mre 17597 df-mrc 17598 df-acs 17600 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18447 df-clat 18514 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-subg 19148 df-cntz 19340 df-oppg 19369 df-lsm 19659 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20542 df-rlreg 20723 df-domn 20724 df-drng 20760 df-lmod 20909 df-lss 20979 df-lsp 21019 df-lvec 21150 df-lsatoms 39564 df-lshyp 39565 df-lcv 39607 df-lfl 39646 df-lkr 39674 df-ldual 39712 df-oposet 39764 df-ol 39766 df-oml 39767 df-covers 39854 df-ats 39855 df-atl 39886 df-cvlat 39910 df-hlat 39939 df-llines 40086 df-lplanes 40087 df-lvols 40088 df-lines 40089 df-psubsp 40091 df-pmap 40092 df-padd 40384 df-lhyp 40576 df-laut 40577 df-ldil 40692 df-ltrn 40693 df-trl 40747 df-tgrp 41331 df-tendo 41343 df-edring 41345 df-dveca 41591 df-disoa 41617 df-dvech 41667 df-dib 41727 df-dic 41761 df-dih 41817 df-doch 41936 df-djh 41983

This theorem is referenced by: lcfrlem38 42168

W3C validator