lcfrlem39 - Mathbox for Norm Megill

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Theorem lcfrlem39 41605

Description: Lemma for lcfr 41609. 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 2736	. 2 ⊢ (Base‘𝑈) = (Base‘𝑈)
25		eqid 2736	. 2 ⊢ ( ·_𝑠 ‘𝑈) = ( ·_𝑠 ‘𝑈)
26		eqid 2736	. 2 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
27		eqid 2736	. 2 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
28		oveq1 7417	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → (𝑗( ·_𝑠 ‘𝑈)𝑥) = (𝑘( ·_𝑠 ‘𝑈)𝑥))
29	28	oveq2d 7426	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)) = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥)))
30	29	eqeq2d 2747	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)) ↔ 𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥))))
31	30	rexbidv 3165	. . . . 5 ⊢ (𝑗 = 𝑘 → (∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑗( ·_𝑠 ‘𝑈)𝑥)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘( ·_𝑠 ‘𝑈)𝑥))))
32	31	cbvriotavw 7377	. . . 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 5222	. . 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 5222	. 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 41604	1 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐸)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∃wrex 3061 {crab 3420 ∖ cdif 3928 ∩ cin 3930 ⊆ wss 3931 {csn 4606 {cpr 4608 ∪ ciun 4972 ↦ cmpt 5206 ‘cfv 6536 ℩crio 7366 (class class class)co 7410 Basecbs 17233 +_gcplusg 17276 Scalarcsca 17279 ·_𝑠 cvsca 17280 0_gc0g 17458 LSubSpclss 20893 LSpanclspn 20933 LFnlclfn 39080 LKerclk 39108 LDualcld 39146 HLchlt 39373 LHypclh 40008 DVecHcdvh 41102 ocHcoch 41371

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 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211 ax-riotaBAD 38976

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 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7676 df-om 7867 df-1st 7993 df-2nd 7994 df-tpos 8230 df-undef 8277 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-struct 17171 df-sets 17188 df-slot 17206 df-ndx 17218 df-base 17234 df-ress 17257 df-plusg 17289 df-mulr 17290 df-sca 17292 df-vsca 17293 df-0g 17460 df-mre 17603 df-mrc 17604 df-acs 17606 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19622 df-cmn 19768 df-abl 19769 df-mgp 20106 df-rng 20118 df-ur 20147 df-ring 20200 df-oppr 20302 df-dvdsr 20322 df-unit 20323 df-invr 20353 df-dvr 20366 df-nzr 20478 df-rlreg 20659 df-domn 20660 df-drng 20696 df-lmod 20824 df-lss 20894 df-lsp 20934 df-lvec 21066 df-lsatoms 38999 df-lshyp 39000 df-lcv 39042 df-lfl 39081 df-lkr 39109 df-ldual 39147 df-oposet 39199 df-ol 39201 df-oml 39202 df-covers 39289 df-ats 39290 df-atl 39321 df-cvlat 39345 df-hlat 39374 df-llines 39522 df-lplanes 39523 df-lvols 39524 df-lines 39525 df-psubsp 39527 df-pmap 39528 df-padd 39820 df-lhyp 40012 df-laut 40013 df-ldil 40128 df-ltrn 40129 df-trl 40183 df-tgrp 40767 df-tendo 40779 df-edring 40781 df-dveca 41027 df-disoa 41053 df-dvech 41103 df-dib 41163 df-dic 41197 df-dih 41253 df-doch 41372 df-djh 41419

This theorem is referenced by: lcfrlem40 41606

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