lcfrlem33 - Mathbox for Norm Megill

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Theorem lcfrlem33 41576

Description: Lemma for lcfr 41586. (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 7406	. . . . . . . 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 41111	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		lcfrlem24.s	. . . . . . . . . . 11 ⊢ 𝑆 = (Scalar‘𝑈)
9	8	lmodring 20781	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring)
10	7, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Ring)
11	4, 5, 6	dvhlvec 41110	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ LVec)
12	8	lvecdrng 21019	. . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
21		lcfrlem24.l	. . . . . . . . . . . 12 ⊢ 𝐿 = (LKer‘𝑈)
22		lcfrlem25.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (LDual‘𝑈)
23		eqid 2730	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
24		eqid 2730	. . . . . . . . . . . 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 41553	. . . . . . . . . . 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 41565	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
34	15, 28, 7, 33	lsatssv 38998	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ⊆ 𝑉)
35		lcfrlem24.ib	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
36	34, 35	sseldd 3950	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
37	8, 18, 15, 20	lflcl 39064	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑌) ∈ (LFnl‘𝑈) ∧ 𝐼 ∈ 𝑉) → ((𝐽‘𝑌)‘𝐼) ∈ 𝑅)
38	7, 27, 36, 37	syl3anc 1373	. . . . . . . . . 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 20669	. . . . . . . . . 10 ⊢ ((𝑆 ∈ DivRing ∧ ((𝐽‘𝑌)‘𝐼) ∈ 𝑅 ∧ ((𝐽‘𝑌)‘𝐼) ≠ 𝑄) → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
43	13, 38, 39, 42	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅)
44		eqid 2730	. . . . . . . . . 10 ⊢ (._r‘𝑆) = (._r‘𝑆)
45	18, 44, 40	ringrz 20210	. . . . . . . . 9 ⊢ ((𝑆 ∈ Ring ∧ (𝐹‘((𝐽‘𝑌)‘𝐼)) ∈ 𝑅) → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)𝑄) = 𝑄)
46	10, 43, 45	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)𝑄) = 𝑄)
47	3, 46	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼)) = 𝑄)
48	47	oveq1d 7405	. . . . . 6 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) = (𝑄( ·_𝑠 ‘𝐷)(𝐽‘𝑌)))
49		eqid 2730	. . . . . . 7 ⊢ ( ·_𝑠 ‘𝐷) = ( ·_𝑠 ‘𝐷)
50	20, 8, 40, 22, 49, 23, 7, 27	ldual0vs 39160	. . . . . 6 ⊢ (𝜑 → (𝑄( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) = (0_g‘𝐷))
51	48, 50	eqtrd 2765	. . . . 5 ⊢ (𝜑 → (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌)) = (0_g‘𝐷))
52	51	oveq2d 7406	. . . 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 39146	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ Grp)
54		eqid 2730	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
55	4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 30	lcfrlem10 41553	. . . . . 6 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LFnl‘𝑈))
56	20, 22, 54, 7, 55	ldualelvbase 39127	. . . . 5 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (Base‘𝐷))
57		lcfrlem30.m	. . . . . 6 ⊢ − = (-_g‘𝐷)
58	54, 23, 57	grpsubid1 18964	. . . . 5 ⊢ ((𝐷 ∈ Grp ∧ (𝐽‘𝑋) ∈ (Base‘𝐷)) → ((𝐽‘𝑋) − (0_g‘𝐷)) = (𝐽‘𝑋))
59	53, 56, 58	syl2anc 584	. . . 4 ⊢ (𝜑 → ((𝐽‘𝑋) − (0_g‘𝐷)) = (𝐽‘𝑋))
60	52, 59	eqtrd 2765	. . 3 ⊢ (𝜑 → ((𝐽‘𝑋) − (((𝐹‘((𝐽‘𝑌)‘𝐼))(._r‘𝑆)((𝐽‘𝑋)‘𝐼))( ·_𝑠 ‘𝐷)(𝐽‘𝑌))) = (𝐽‘𝑋))
61	1, 60	eqtrid 2777	. 2 ⊢ (𝜑 → 𝐶 = (𝐽‘𝑋))
62	4, 14, 5, 15, 16, 17, 8, 18, 19, 20, 21, 22, 23, 24, 25, 6, 30	lcfrlem13 41556	. . 3 ⊢ (𝜑 → (𝐽‘𝑋) ∈ ({𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ∖ {(0_g‘𝐷)}))
63		eldifsni 4757	. . 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 2993	1 ⊢ (𝜑 → 𝐶 ≠ (0_g‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∃wrex 3054 {crab 3408 ∖ cdif 3914 ∩ cin 3916 {csn 4592 {cpr 4594 ↦ cmpt 5191 ‘cfv 6514 ℩crio 7346 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 ._rcmulr 17228 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17409 Grpcgrp 18872 -_gcsg 18874 Ringcrg 20149 inv_rcinvr 20303 DivRingcdr 20645 LModclmod 20773 LSpanclspn 20884 LVecclvec 21016 LSAtomsclsa 38974 LFnlclfn 39057 LKerclk 39085 LDualcld 39123 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 ocHcoch 41348

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

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 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396

This theorem is referenced by: lcfrlem34 41577

W3C validator