lcfrlem39 - Mathbox for Norm Megill

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Theorem lcfrlem39 36870

Description: Lemma for lcfr 36874. Eliminate 𝐽. (Contributed by NM, 11-Mar-2015.)

**Hypotheses**
Ref	Expression
lcfrlem38.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfrlem38.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfrlem38.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem38.p	⊢ + = (+_g‘𝑈)
lcfrlem38.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfrlem38.l	⊢ 𝐿 = (LKer‘𝑈)
lcfrlem38.d	⊢ 𝐷 = (LDual‘𝑈)
lcfrlem38.q	⊢ 𝑄 = (LSubSp‘𝐷)
lcfrlem38.c	⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcfrlem38.e	⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
lcfrlem38.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem38.g	⊢ (𝜑 → 𝐺 ∈ 𝑄)
lcfrlem38.gs	⊢ (𝜑 → 𝐺 ⊆ 𝐶)
lcfrlem38.xe	⊢ (𝜑 → 𝑋 ∈ 𝐸)
lcfrlem38.ye	⊢ (𝜑 → 𝑌 ∈ 𝐸)
lcfrlem38.z	⊢ 0 = (0_g‘𝑈)
lcfrlem38.x	⊢ (𝜑 → 𝑋 ≠ 0 )
lcfrlem38.y	⊢ (𝜑 → 𝑌 ≠ 0 )
lcfrlem38.sp	⊢ 𝑁 = (LSpan‘𝑈)
lcfrlem38.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem38.b	⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
lcfrlem38.i	⊢ (𝜑 → 𝐼 ∈ 𝐵)
lcfrlem38.n	⊢ (𝜑 → 𝐼 ≠ 0 )

**Assertion**
Ref	Expression
lcfrlem39	⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Distinct variable groups: 𝐷,𝑔 𝑔,𝐺 𝑔,𝐼 𝑓,𝑔,𝐿 ⊥ ,𝑓,𝑔 + ,𝑓,𝑔 𝑈,𝑓,𝑔 𝑓,𝑋,𝑔 𝑓,𝑌,𝑔 0 ,𝑓,𝑔 𝜑,𝑔 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓,𝑔) 𝐶(𝑓,𝑔) 𝐷(𝑓) 𝑄(𝑓,𝑔) 𝐸(𝑓,𝑔) 𝐹(𝑓,𝑔) 𝐺(𝑓) 𝐻(𝑓,𝑔) 𝐼(𝑓) 𝐾(𝑓,𝑔) 𝑁(𝑓,𝑔) 𝑊(𝑓,𝑔)

Proof of Theorem lcfrlem39 Dummy variables 𝑘 𝑗 𝑣 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcfrlem38.h	. 2 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfrlem38.o	. 2 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcfrlem38.u	. 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcfrlem38.p	. 2 ⊢ + = (+_g‘𝑈)
5		lcfrlem38.f	. 2 ⊢ 𝐹 = (LFnl‘𝑈)
6		lcfrlem38.l	. 2 ⊢ 𝐿 = (LKer‘𝑈)
7		lcfrlem38.d	. 2 ⊢ 𝐷 = (LDual‘𝑈)
8		lcfrlem38.q	. 2 ⊢ 𝑄 = (LSubSp‘𝐷)
9		lcfrlem38.c	. 2 ⊢ 𝐶 = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
10		lcfrlem38.e	. 2 ⊢ 𝐸 = ∪ 𝑔 ∈ 𝐺 ( ⊥ ‘(𝐿‘𝑔))
11		lcfrlem38.k	. 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		lcfrlem38.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑄)
13		lcfrlem38.gs	. 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐶)
14		lcfrlem38.xe	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐸)
15		lcfrlem38.ye	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐸)
16		lcfrlem38.z	. 2 ⊢ 0 = (0_g‘𝑈)
17		lcfrlem38.x	. 2 ⊢ (𝜑 → 𝑋 ≠ 0 )
18		lcfrlem38.y	. 2 ⊢ (𝜑 → 𝑌 ≠ 0 )
19		lcfrlem38.sp	. 2 ⊢ 𝑁 = (LSpan‘𝑈)
20		lcfrlem38.ne	. 2 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
21		lcfrlem38.b	. 2 ⊢ 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ⊥ ‘{(𝑋 + 𝑌)}))
22		lcfrlem38.i	. 2 ⊢ (𝜑 → 𝐼 ∈ 𝐵)
23		lcfrlem38.n	. 2 ⊢ (𝜑 → 𝐼 ≠ 0 )
24		eqid 2622	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
25		eqid 2622	. 2 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
26		eqid 2622	. 2 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
27		eqid 2622	. 2 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
28		oveq1 6657	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗( ·_𝑠 ‘𝑈)𝑥) = (𝑘( ·_𝑠 ‘𝑈)𝑥))
29	28	oveq2d 6666	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)) = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥)))
30	29	eqeq2d 2632	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)) ↔ 𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥))))
31	30	rexbidv 3052	. . . . 5 ⊢ (𝑗 = 𝑘 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥))))
32	31	cbvriotav 6622	. . . 4 ⊢ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥))) = (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥)))
33	32	mpteq2i 4741	. . 3 ⊢ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)))) = (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥))))
34	33	mpteq2i 4741	. 2 ⊢ (𝑥 ∈ ((Base‘𝑈) ∖ { 0 }) ↦ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑗 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥))))) = (𝑥 ∈ ((Base‘𝑈) ∖ { 0 }) ↦ (𝑣 ∈ (Base‘𝑈) ↦ (℩𝑘 ∈ (Base‘(Scalar‘𝑈))∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥)))))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 34	lcfrlem38 36869	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 {crab 2916 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 {csn 4177 {cpr 4179 ∪ ciun 4520 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 LSubSpclss 18932 LSpanclspn 18971 LFnlclfn 34344 LKerclk 34372 LDualcld 34410 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 ocHcoch 36636

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684

This theorem is referenced by: lcfrlem40 36871

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