lcfrlem33 - Mathbox for Norm Megill

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Theorem lcfrlem33 42163

Description: Lemma for lcfr 42173. (Contributed by NM, 10-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)))
lcfrlem33.xi	⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) = 𝑄)

**Assertion**
Ref	Expression
lcfrlem33	⊢ (𝜑 → 𝐶 ≠ (0_g‘𝐷))

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 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑘) 𝐴(𝑥,𝑤,𝑣,𝑘) 𝐵(𝑥,𝑤,𝑣,𝑘) 𝐶(𝑥,𝑤,𝑣,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑘) 𝑄(𝑥,𝑤,𝑣,𝑘) 𝑅(𝑤) 𝑆(𝑥,𝑤,𝑣) 𝑈(𝑥,𝑤,𝑣,𝑘) 𝐹(𝑥,𝑤,𝑣,𝑘) 𝐻(𝑥,𝑤,𝑣,𝑘) 𝐼(𝑥,𝑤,𝑣,𝑘) 𝐽(𝑥,𝑤,𝑣,𝑘) 𝐾(𝑥,𝑤,𝑣,𝑘) 𝐿(𝑥,𝑤,𝑣,𝑘) − (𝑥,𝑤,𝑣,𝑘) 𝑁(𝑥,𝑤,𝑣,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑘) 0 (𝑤,𝑣,𝑘)

Proof of Theorem lcfrlem33 Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem30.c	. . 3 ⊢ 𝐶 = ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
2		lcfrlem33.xi	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝐼) = 𝑄)
3	2	oveq2d 7408	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) = ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)𝑄))
4		lcfrlem17.h	. . . . . . . . . . 11 ⊢ 𝐻 = (LHyp‘𝐾)
5		lcfrlem17.u	. . . . . . . . . . 11 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lcfrlem17.k	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	dvhlmod 41698	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		lcfrlem24.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑈)
9	8	lmodring 20915	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring)
11	4, 5, 6	dvhlvec 41697	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LVec)
12	8	lvecdrng 21152	. . . . . . . . . . 11 ⊢ (𝑈 ∈ LVec → 𝑆 ∈ DivRing)
13	11, 12	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ DivRing)
14		lcfrlem17.o	. . . . . . . . . . . 12 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15		lcfrlem17.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Base‘𝑈)
16		lcfrlem17.p	. . . . . . . . . . . 12 ⊢ + = (+_g‘𝑈)
17		lcfrlem24.t	. . . . . . . . . . . 12 ⊢ · = ( ·_𝑠 ‘𝑈)
18		lcfrlem24.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Base‘𝑆)
19		lcfrlem17.z	. . . . . . . . . . . 12 ⊢ 0 = (0_g‘𝑈)
20		eqid 2761	. . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
21		lcfrlem24.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LKer‘𝑈)
22		lcfrlem25.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (LDual‘𝑈)
23		eqid 2761	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
24		eqid 2761	. . . . . . . . . . . 12 ⊢ {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
25		lcfrlem24.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
26		lcfrlem17.y	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
27	4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 26	lcfrlem10 42140	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LFnl‘𝑈))
28		lcfrlem17.a	. . . . . . . . . . . . 13 ⊢ 𝐴 = (LSAtoms‘𝑈)
29		lcfrlem17.n	. . . . . . . . . . . . . 14 ⊢ 𝑁 = (LSpan‘𝑈)
30		lcfrlem17.x	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
31		lcfrlem17.ne	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
32		lcfrlem22.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
33	4, 14, 5, 15, 16, 19, 29, 28, 6, 30, 26, 31, 32	lcfrlem22 42152	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
34	15, 28, 7, 33	lsatssv 39586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
35		lcfrlem24.ib	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
36	34, 35	sseldd 3937	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
37	8, 18, 15, 20	lflcl 39652	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
38	7, 27, 36, 37	syl3anc 1389	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
39		lcfrlem28.jn	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐽‘𝑌)‘𝐼) ≠ 𝑄)
40		lcfrlem24.q	. . . . . . . . . . 11 ⊢ 𝑄 = (0_g‘𝑆)
41		lcfrlem29.i	. . . . . . . . . . 11 ⊢ 𝐹 = (inv_r‘𝑆)
42	18, 40, 41	drnginvrcl 20782	. . . . . . . . . 10 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
43	13, 38, 39, 42	syl3anc 1389	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
44		eqid 2761	. . . . . . . . . 10 ⊢ (._r‘𝑆) = (._r‘𝑆)
45	18, 44, 40	ringrz 20323	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅) → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)𝑄) = 𝑄)
46	10, 43, 45	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)𝑄) = 𝑄)
47	3, 46	eqtrd 2796	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) = 𝑄)
48	47	oveq1d 7407	. . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) = (𝑄( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
49		eqid 2761	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
50	20, 8, 40, 22, 49, 23, 7, 27	ldual0vs 39748	. . . . . 6 ⊢ (𝜑 → (𝑄( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) = (0_g‘𝐷))
51	48, 50	eqtrd 2796	. . . . 5 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) = (0_g‘𝐷))
52	51	oveq2d 7408	. . . 4 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) = ((𝐽‘𝑋) − (0_g‘𝐷)))
53	22, 7	ldualgrp 39734	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Grp)
54		eqid 2761	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
55	4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 30	lcfrlem10 42140	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
56	20, 22, 54, 7, 55	ldualelvbase 39715	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷))
57		lcfrlem30.m	. . . . . 6 ⊢ − = (-_g‘𝐷)
58	54, 23, 57	grpsubid1 19050	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (𝐽‘𝑋) ∈ (Base‘𝐷)) → ((𝐽‘𝑋) − (0_g‘𝐷)) = (𝐽‘𝑋))
59	53, 56, 58	syl2anc 593	. . . 4 ⊢ (𝜑 → ((𝐽‘𝑋) − (0_g‘𝐷)) = (𝐽‘𝑋))
60	52, 59	eqtrd 2796	. . 3 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐽‘𝑋))
61	1, 60	eqtrid 2808	. 2 ⊢ (𝜑 → 𝐶 = (𝐽‘𝑋))
62	4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 30	lcfrlem13 42143	. . 3 ⊢ (𝜑 → (𝐽‘𝑋) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0_g‘𝐷)}))
63		eldifsni 4749	. . 3 ⊢ ((𝐽‘𝑋) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0_g‘𝐷)}) → (𝐽‘𝑋) ≠ (0_g‘𝐷))
64	62, 63	syl 17	. 2 ⊢ (𝜑 → (𝐽‘𝑋) ≠ (0_g‘𝐷))
65	61, 64	eqnetrd 3023	1 ⊢ (𝜑 → 𝐶 ≠ (0_g‘𝐷))

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 {csn 4581 {cpr 4583 ↦ 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 Grpcgrp 18958 -_gcsg 18960 Ringcrg 20262 inv_rcinvr 20415 DivRingcdr 20758 LModclmod 20907 LSpanclspn 21018 LVecclvec 21149 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-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: lcfrlem34 42164

W3C validator