Theorem dib1dim 41801

Description: Two expressions for the 1-dimensional subspaces of vector space H. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
dib1dim.b	⊢ 𝐵 = (Base‘𝐾)
dib1dim.h	⊢ 𝐻 = (LHyp‘𝐾)
dib1dim.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dib1dim.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dib1dim.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dib1dim.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dib1dim.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)

Assertion

Ref	Expression
dib1dim	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩})

Distinct variable groups:   𝐵,ℎ   𝑔,𝑠,𝐸   𝑔,𝐹,𝑠   𝐻,𝑠   ℎ,𝑠,𝐾   𝑔,𝑂,𝑠   𝑅,𝑠   𝑔,ℎ,𝑇,𝑠   ℎ,𝑊,𝑠

Allowed substitution hints:   𝐵(𝑔,𝑠)   𝑅(𝑔,ℎ)   𝐸(ℎ)   𝐹(ℎ)   𝐻(𝑔,ℎ)   𝐼(𝑔,ℎ,𝑠)   𝐾(𝑔)   𝑂(ℎ)   𝑊(𝑔)

Proof of Theorem dib1dim

Dummy variables 𝑓 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 487	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dib1dim.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
3		dib1dim.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		dib1dim.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		dib1dim.r	. . . . 5 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
6	2, 3, 4, 5	trlcl 40800	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵)
7		eqid 2765	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
8	7, 3, 4, 5	trlle 40820	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)𝑊)
9		dib1dim.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
10		eqid 2765	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
11		dib1dim.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
12	2, 7, 3, 4, 9, 10, 11	dibval2 41780	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅‘𝐹) ∈ 𝐵 ∧ (𝑅‘𝐹)(le‘𝐾)𝑊)) → (𝐼‘(𝑅‘𝐹)) = ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂}))
13	1, 6, 8, 12	syl12anc 849	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂}))
14		relxp 5670	. . . 4 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂})
15		opelxp 5688	. . . . 5 ⊢ (⟨𝑓, 𝑡⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂}) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) ∧ 𝑡 ∈ {𝑂}))
16		dib1dim.e	. . . . . . . . 9 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
17	3, 4, 5, 16, 10	dia1dim 41697	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) = {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)})
18	17	eqabrd 2906	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) ↔ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)))
19	18	anbi1d 642	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) ∧ 𝑡 ∈ {𝑂}) ↔ (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 ∈ {𝑂})))
20	3, 4, 16	tendocl 41403	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
21	20	3expa 1134	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
22	21	an32s 664	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → (𝑠‘𝐹) ∈ 𝑇)
23	2, 3, 4, 16, 9	tendo0cl 41426	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
24	23	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → 𝑂 ∈ 𝐸)
25	22, 24	jca 520	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑠‘𝐹) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸))
26		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠‘𝐹) → (𝑓 ∈ 𝑇 ↔ (𝑠‘𝐹) ∈ 𝑇))
27		eleq1 2853	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑂 → (𝑡 ∈ 𝐸 ↔ 𝑂 ∈ 𝐸))
28	26, 27	bi2anan9 649	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) → ((𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ↔ ((𝑠‘𝐹) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸)))
29	25, 28	syl5ibrcom 250	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) → (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)))
30	29	rexlimdva 3166	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) → (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)))
31	30	pm4.71rd 571	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) ↔ ((𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))))
32		velsn 4601	. . . . . . . . 9 ⊢ (𝑡 ∈ {𝑂} ↔ 𝑡 = 𝑂)
33	32	anbi2i 634	. . . . . . . 8 ⊢ ((∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
34		r19.41v 3195	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) ↔ (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
35	33, 34	bitr4i 281	. . . . . . 7 ⊢ ((∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
36		df-3an 1103	. . . . . . 7 ⊢ ((𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂)) ↔ ((𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂)))
37	31, 35, 36	3bitr4g 317	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))))
38	19, 37	bitrd 282	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) ∧ 𝑡 ∈ {𝑂}) ↔ (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))))
39	15, 38	bitrid 286	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (⟨𝑓, 𝑡⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂}) ↔ (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))))
40	14, 39	opabbi2dv 5826	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂}) = {⟨𝑓, 𝑡⟩ ∣ (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))})
41	13, 40	eqtrd 2800	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = {⟨𝑓, 𝑡⟩ ∣ (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))})
42		eqeq1 2769	. . . . 5 ⊢ (𝑔 = ⟨𝑓, 𝑡⟩ → (𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ ⟨𝑓, 𝑡⟩ = ⟨(𝑠‘𝐹), 𝑂⟩))
43		vex 3461	. . . . . 6 ⊢ 𝑓 ∈ V
44		vex 3461	. . . . . 6 ⊢ 𝑡 ∈ V
45	43, 44	opth 5449	. . . . 5 ⊢ (⟨𝑓, 𝑡⟩ = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
46	42, 45	bitrdi 290	. . . 4 ⊢ (𝑔 = ⟨𝑓, 𝑡⟩ → (𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂)))
47	46	rexbidv 3189	. . 3 ⊢ (𝑔 = ⟨𝑓, 𝑡⟩ → (∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂)))
48	47	rabxp 5700	. 2 ⊢ {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩} = {⟨𝑓, 𝑡⟩ ∣ (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸 ∧ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))}
49	41, 48	eqtr4di 2818	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩})

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  {crab 3417  {csn 4585  ⟨cop 4591   class class class wbr 5105  {copab 5167   ↦ cmpt 5186   I cid 5546   × cxp 5650   ↾ cres 5654  ‘cfv 6525  Basecbs 17259  lecple 17307  HLchlt 39986  LHypclh 40620  LTrncltrn 40737  trLctrl 40794  TEndoctendo 41388  DIsoAcdia 41664  DIsoBcdib 41774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-riotaBAD 39589

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-undef 8257  df-map 8814  df-proset 18340  df-poset 18359  df-plt 18374  df-lub 18390  df-glb 18391  df-join 18392  df-meet 18393  df-p0 18469  df-p1 18470  df-lat 18478  df-clat 18545  df-oposet 39812  df-ol 39814  df-oml 39815  df-covers 39902  df-ats 39903  df-atl 39934  df-cvlat 39958  df-hlat 39987  df-llines 40134  df-lplanes 40135  df-lvols 40136  df-lines 40137  df-psubsp 40139  df-pmap 40140  df-padd 40432  df-lhyp 40624  df-laut 40625  df-ldil 40740  df-ltrn 40741  df-trl 40795  df-tendo 41391  df-disoa 41665  df-dib 41775

This theorem is referenced by:  dib1dim2  41804