dochfl1 - Mathbox for Norm Megill

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Theorem dochfl1 42535

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 3909	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		eqeq1 2765	. . . . . 6 ⊢ (𝑣 = 𝑋 → (𝑣 = (𝑤 + (𝑘 · 𝑋)) ↔ 𝑋 = (𝑤 + (𝑘 · 𝑋))))
4	3	rexbidv 3187	. . . . 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 42169	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
14	2	snssd 4747	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
15		dochfl1.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
16		eqid 2761	. . . . . . 7 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
17		dochfl1.o	. . . . . . 7 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18	10, 11, 15, 16, 17	dochlss 42413	. . . . . 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 21223	. . . . 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 21166	. . . . . . 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 21168	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ( 0 + 𝑋) = 𝑋)
31	13, 2, 30	syl2anc 596	. . . . 5 ⊢ (𝜑 → ( 0 + 𝑋) = 𝑋)
32	28, 31	eqtr2d 2797	. . . 4 ⊢ (𝜑 → 𝑋 = ( 0 + ( 1 · 𝑋)))
33		oveq1 7428	. . . . 5 ⊢ (𝑤 = 0 → (𝑤 + ( 1 · 𝑋)) = ( 0 + ( 1 · 𝑋)))
34	33	rspceeqv 3599	. . . 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 21144	. . . . 5 ⊢ (𝑈 ∈ LMod → 𝐷 ∈ Ring)
37		dochfl1.r	. . . . . 6 ⊢ 𝑅 = (Base‘𝐷)
38	37, 25	ringidcl 20495	. . . . 5 ⊢ (𝐷 ∈ Ring → 1 ∈ 𝑅)
39	13, 36, 38	3syl 19	. . . 4 ⊢ (𝜑 → 1 ∈ 𝑅)
40		eqid 2761	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
41		eqid 2761	. . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
42		eqid 2761	. . . . 5 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
43	10, 11, 12	dvhlvec 42168	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec)
44	10, 17, 11, 15, 20, 42, 12, 1	dochsnshp 42512	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈))
45	10, 17, 11, 15, 20, 40, 41, 12, 1	dochexmidat 42518	. . . . 5 ⊢ (𝜑 → (( ⊥ ‘{𝑋})(LSSum‘𝑈)((LSpan‘𝑈)‘{𝑋})) = 𝑉)
46	15, 29, 40, 41, 42, 43, 44, 2, 2, 45, 23, 37, 24	lshpsmreu 40168	. . . 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 2772	. . . . . 6 ⊢ (𝑘 = 1 → (𝑋 = (𝑤 + (𝑘 · 𝑋)) ↔ 𝑋 = (𝑤 + ( 1 · 𝑋))))
50	49	rexbidv 3187	. . . . 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 2796	1 ⊢ (𝜑 → (𝐺‘𝑋) = 1 )

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∃wrex 3087 ∃!wreu 3364 ∖ cdif 3896 ⊆ wss 3914 {csn 4584 ↦ cmpt 5186 ‘cfv 6539 ℩crio 7377 (class class class)co 7421 Basecbs 17388 +_gcplusg 17429 Scalarcsca 17432 ·_𝑠 cvsca 17433 0_gc0g 17611 LSSumclsm 19849 1_rcur 20408 Ringcrg 20460 LModclmod 21136 LSubSpclss 21207 LSpanclspn 21247 LSHypclsh 40034 HLchlt 40409 LHypclh 41043 DVecHcdvh 42137 ocHcoch 42406

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7752 ax-cnex 11257 ax-resscn 11258 ax-1cn 11259 ax-icn 11260 ax-addcl 11261 ax-addrcl 11262 ax-mulcl 11263 ax-mulrcl 11264 ax-mulcom 11265 ax-addass 11266 ax-mulass 11267 ax-distr 11268 ax-i2m1 11269 ax-1ne0 11270 ax-1rid 11271 ax-rnegex 11272 ax-rrecex 11273 ax-cnre 11274 ax-pre-lttri 11275 ax-pre-lttrn 11276 ax-pre-ltadd 11277 ax-pre-mulgt0 11278 ax-riotaBAD 40012

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3900 df-un 3902 df-in 3904 df-ss 3915 df-pss 4035 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 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 8244 df-undef 8291 df-frecs 8300 df-wrecs 8331 df-recs 8380 df-rdg 8419 df-1o 8477 df-er 8718 df-map 8850 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11346 df-mnf 11347 df-xr 11348 df-ltxr 11349 df-le 11350 df-sub 11544 df-neg 11545 df-nn 12337 df-2 12406 df-3 12407 df-4 12408 df-5 12409 df-6 12410 df-n0 12608 df-z 12695 df-uz 12967 df-fz 13641 df-struct 17326 df-sets 17343 df-slot 17361 df-ndx 17373 df-base 17389 df-ress 17410 df-plusg 17442 df-mulr 17443 df-sca 17445 df-vsca 17446 df-0g 17613 df-proset 18469 df-poset 18488 df-plt 18503 df-lub 18519 df-glb 18520 df-join 18521 df-meet 18522 df-p0 18598 df-p1 18599 df-lat 18607 df-clat 18674 df-mgm 18817 df-sgrp 18909 df-mnd 18925 df-submnd 18980 df-grp 19148 df-minusg 19149 df-sbg 19150 df-subg 19334 df-cntz 19532 df-lsm 19851 df-cmn 19997 df-abl 19998 df-mgp 20362 df-rng 20376 df-ur 20409 df-ring 20462 df-oppr 20568 df-dvdsr 20588 df-unit 20589 df-invr 20619 df-dvr 20632 df-drng 20983 df-lmod 21138 df-lss 21208 df-lsp 21248 df-lvec 21379 df-lsatoms 40035 df-lshyp 40036 df-oposet 40235 df-ol 40237 df-oml 40238 df-covers 40325 df-ats 40326 df-atl 40357 df-cvlat 40381 df-hlat 40410 df-llines 40557 df-lplanes 40558 df-lvols 40559 df-lines 40560 df-psubsp 40562 df-pmap 40563 df-padd 40855 df-lhyp 41047 df-laut 41048 df-ldil 41163 df-ltrn 41164 df-trl 41218 df-tgrp 41802 df-tendo 41814 df-edring 41816 df-dveca 42062 df-disoa 42088 df-dvech 42138 df-dib 42198 df-dic 42232 df-dih 42288 df-doch 42407 df-djh 42454

This theorem is used by: lcfl6lem 42557 lcfl7lem 42558 hvmapidN 42821 hdmapevec2 42895

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