dochfl1 - Mathbox for Norm Megill

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Theorem dochfl1 42627

Description: The value of the explicit functional 𝐺 is 1 at the 𝑋 that determines it. (Contributed by NM, 27-Oct-2014.)

**Hypotheses**
Ref	Expression
dochfl1.h	⊢ 𝐻 = (LHyp‘𝐾)
dochfl1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochfl1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochfl1.v	⊢ 𝑉 = (Base‘𝑈)
dochfl1.a	⊢ + = (+_g‘𝑈)
dochfl1.t	⊢ · = ( ·_𝑠 ‘𝑈)
dochfl1.z	⊢ 0 = (0_g‘𝑈)
dochfl1.d	⊢ 𝐷 = (Scalar‘𝑈)
dochfl1.r	⊢ 𝑅 = (Base‘𝐷)
dochfl1.i	⊢ 1 = (1_r‘𝐷)
dochfl1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochfl1.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
dochfl1.g	⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))

**Assertion**
Ref	Expression
dochfl1	⊢ (𝜑 → (𝐺‘𝑋) = 1 )

Distinct variable groups: 𝑣,𝑘,𝑤, + 1 ,𝑘,𝑤 ⊥ ,𝑘,𝑣,𝑤 𝑅,𝑘,𝑣 · ,𝑘,𝑣,𝑤 𝑣,𝑉 𝑘,𝑋,𝑣,𝑤 𝑤, 0 Allowed substitution hints: 𝜑(𝑤, 𝑣, 𝑘) 𝐷(𝑤, 𝑣, 𝑘) 𝑅(𝑤) 𝑈(𝑤, 𝑣, 𝑘) 1 (𝑣) 𝐺(𝑤, 𝑣, 𝑘) 𝐻(𝑤, 𝑣, 𝑘) 𝐾(𝑤, 𝑣, 𝑘) 𝑉(𝑤, 𝑘) 𝑊(𝑤, 𝑣, 𝑘) 0 (𝑣, 𝑘)

**Proof of Theorem dochfl1**
Step	Hyp	Ref	Expression
1		dochfl1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
2	1	eldifad 4051	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		eqeq1 2764	. . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑋)) ↔ 𝑋 = (𝑤 + (𝑘 · 𝑋))))
4	3	rexbidv 3186	. . . . 5 ⊢ (𝑣 = 𝑋 → (∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))))
5	4	riotabidv 7380	. . . 4 ⊢ (𝑣 = 𝑋 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))))
6		dochfl1.g	. . . 4 ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))
7		riotaex 7382	. . . 4 ⊢ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))) ∈ V
8	5, 6, 7	fvmpt 6994	. . 3 ⊢ (𝑋 ∈ 𝑉 → (𝐺‘𝑋) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))))
9	2, 8	syl 18	. 2 ⊢ (𝜑 → (𝐺‘𝑋) = (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))))
10		dochfl1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
11		dochfl1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
12		dochfl1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13	10, 11, 12	dvhlmod 42261	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	2	snssd 4743	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
15		dochfl1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
16		eqid 2760	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
17		dochfl1.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18	10, 11, 15, 16, 17	dochlss 42505	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ (LSubSp‘𝑈))
19	12, 14, 18	syl2anc 596	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ (LSubSp‘𝑈))
20		dochfl1.z	. . . . . 6 ⊢ 0 = (0_g‘𝑈)
21	20, 16	lss0cl 21228	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ ( ⊥ ‘{𝑋}) ∈ (LSubSp‘𝑈)) → 0 ∈ ( ⊥ ‘{𝑋}))
22	13, 19, 21	syl2anc 596	. . . 4 ⊢ (𝜑 → 0 ∈ ( ⊥ ‘{𝑋}))
23		dochfl1.d	. . . . . . . 8 ⊢ 𝐷 = (Scalar‘𝑈)
24		dochfl1.t	. . . . . . . 8 ⊢ · = ( ·_𝑠 ‘𝑈)
25		dochfl1.i	. . . . . . . 8 ⊢ 1 = (1_r‘𝐷)
26	15, 23, 24, 25	lmodvs1 21171	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 1 · 𝑋) = 𝑋)
27	13, 2, 26	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ( 1 · 𝑋) = 𝑋)
28	27	oveq2d 7437	. . . . 5 ⊢ (𝜑 → ( 0 + ( 1 · 𝑋)) = ( 0 + 𝑋))
29		dochfl1.a	. . . . . . 7 ⊢ + = (+_g‘𝑈)
30	15, 29, 20	lmod0vlid 21173	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋)
31	13, 2, 30	syl2anc 596	. . . . 5 ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)
32	28, 31	eqtr2d 2796	. . . 4 ⊢ (𝜑 → 𝑋 = ( 0 + ( 1 · 𝑋)))
33		oveq1 7428	. . . . 5 ⊢ (𝑤 = 0 → (𝑤 + ( 1 · 𝑋)) = ( 0 + ( 1 · 𝑋)))
34	33	rspceeqv 3598	. . . 4 ⊢ (( 0 ∈ ( ⊥ ‘{𝑋}) ∧ 𝑋 = ( 0 + ( 1 · 𝑋))) → ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + ( 1 · 𝑋)))
35	22, 32, 34	syl2anc 596	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + ( 1 · 𝑋)))
36	23	lmodring 21149	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝐷 ∈ Ring)
37		dochfl1.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝐷)
38	37, 25	ringidcl 20500	. . . . 5 ⊢ (𝐷 ∈ Ring → 1 ∈ 𝑅)
39	13, 36, 38	3syl 19	. . . 4 ⊢ (𝜑 → 1 ∈ 𝑅)
40		eqid 2760	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
41		eqid 2760	. . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
42		eqid 2760	. . . . 5 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
43	10, 11, 12	dvhlvec 42260	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
44	10, 17, 11, 15, 20, 42, 12, 1	dochsnshp 42604	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈))
45	10, 17, 11, 15, 20, 40, 41, 12, 1	dochexmidat 42610	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘{𝑋})(LSSum‘𝑈)((LSpan‘𝑈)‘{𝑋})) = 𝑉)
46	15, 29, 40, 41, 42, 43, 44, 2, 2, 45, 23, 37, 24	lshpsmreu 40260	. . . 4 ⊢ (𝜑 → ∃!𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋)))
47		oveq1 7428	. . . . . . . 8 ⊢ (𝑘 = 1 → (𝑘 · 𝑋) = ( 1 · 𝑋))
48	47	oveq2d 7437	. . . . . . 7 ⊢ (𝑘 = 1 → (𝑤 + (𝑘 · 𝑋)) = (𝑤 + ( 1 · 𝑋)))
49	48	eqeq2d 2771	. . . . . 6 ⊢ (𝑘 = 1 → (𝑋 = (𝑤 + (𝑘 · 𝑋)) ↔ 𝑋 = (𝑤 + ( 1 · 𝑋))))
50	49	rexbidv 3186	. . . . 5 ⊢ (𝑘 = 1 → (∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋)) ↔ ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + ( 1 · 𝑋))))
51	50	riota2 7403	. . . 4 ⊢ (( 1 ∈ 𝑅 ∧ ∃!𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))) → (∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + ( 1 · 𝑋)) ↔ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))) = 1 ))
52	39, 46, 51	syl2anc 596	. . 3 ⊢ (𝜑 → (∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + ( 1 · 𝑋)) ↔ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))) = 1 ))
53	35, 52	mpbid 235	. 2 ⊢ (𝜑 → (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑋 = (𝑤 + (𝑘 · 𝑋))) = 1 )
54	9, 53	eqtrd 2795	1 ⊢ (𝜑 → (𝐺‘𝑋) = 1 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ∃wrex 3086 ∃!wreu 3363 ⊆ wss 3895 ∖ cdif 4045 {csn 4584 ↦ cmpt 5185 ‘cfv 6539 ℩crio 7377 (class class class)co 7421 Basecbs 17393 +_gcplusg 17434 Scalarcsca 17437 ·_𝑠 cvsca 17438 0_gc0g 17616 LSSumclsm 19854 1_rcur 20413 Ringcrg 20465 LModclmod 21141 LSubSpclss 21212 LSpanclspn 21252 LSHypclsh 40126 HLchlt 40501 LHypclh 41135 DVecHcdvh 42229 ocHcoch 42498

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281 ax-riotaBAD 40104

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-tpos 8247 df-undef 8294 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-map 8853 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-nn 12340 df-2 12409 df-3 12410 df-4 12411 df-5 12412 df-6 12413 df-n0 12611 df-z 12698 df-uz 12970 df-fz 13644 df-struct 17331 df-sets 17348 df-slot 17366 df-ndx 17378 df-base 17394 df-ress 17415 df-plusg 17447 df-mulr 17448 df-sca 17450 df-vsca 17451 df-0g 17618 df-proset 18474 df-poset 18493 df-plt 18508 df-lub 18524 df-glb 18525 df-join 18526 df-meet 18527 df-p0 18603 df-p1 18604 df-lat 18612 df-clat 18679 df-mgm 18822 df-sgrp 18914 df-mnd 18930 df-submnd 18985 df-grp 19153 df-minusg 19154 df-sbg 19155 df-subg 19339 df-cntz 19537 df-lsm 19856 df-cmn 20002 df-abl 20003 df-mgp 20367 df-rng 20381 df-ur 20414 df-ring 20467 df-oppr 20573 df-dvdsr 20593 df-unit 20594 df-invr 20624 df-dvr 20637 df-drng 20988 df-lmod 21143 df-lss 21213 df-lsp 21253 df-lvec 21384 df-lsatoms 40127 df-lshyp 40128 df-oposet 40327 df-ol 40329 df-oml 40330 df-covers 40417 df-ats 40418 df-atl 40449 df-cvlat 40473 df-hlat 40502 df-llines 40649 df-lplanes 40650 df-lvols 40651 df-lines 40652 df-psubsp 40654 df-pmap 40655 df-padd 40947 df-lhyp 41139 df-laut 41140 df-ldil 41255 df-ltrn 41256 df-trl 41310 df-tgrp 41894 df-tendo 41906 df-edring 41908 df-dveca 42154 df-disoa 42180 df-dvech 42230 df-dib 42290 df-dic 42324 df-dih 42380 df-doch 42499 df-djh 42546

This theorem is used by: lcfl6lem 42649 lcfl7lem 42650 hvmapidN 42913 hdmapevec2 42987

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