Theorem dib1dim 41737

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 485	. . . 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 40736	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹) ∈ 𝐵)
7		eqid 2756	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
8	7, 3, 4, 5	trlle 40756	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑅‘𝐹)(le‘𝐾)𝑊)
9		dib1dim.o	. . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
10		eqid 2756	. . . . 5 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
11		dib1dim.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
12	2, 7, 3, 4, 9, 10, 11	dibval2 41716	. . . 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 845	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂}))
14		relxp 5658	. . . 4 ⊢ Rel ((((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) × {𝑂})
15		opelxp 5676	. . . . 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 41633	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) = {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)})
18	17	eqabrd 2897	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅‘𝐹)) ↔ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)))
19	18	anbi1d 639	. . . . . 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 41339	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
21	20	3expa 1127	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
22	21	an32s 660	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → (𝑠‘𝐹) ∈ 𝑇)
23	2, 3, 4, 16, 9	tendo0cl 41362	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
24	23	ad2antrr 734	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → 𝑂 ∈ 𝐸)
25	22, 24	jca 518	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑠‘𝐹) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸))
26		eleq1 2844	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑠‘𝐹) → (𝑓 ∈ 𝑇 ↔ (𝑠‘𝐹) ∈ 𝑇))
27		eleq1 2844	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑂 → (𝑡 ∈ 𝐸 ↔ 𝑂 ∈ 𝐸))
28	26, 27	bi2anan9 646	. . . . . . . . . 10 ⊢ ((𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) → ((𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸) ↔ ((𝑠‘𝐹) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸)))
29	25, 28	syl5ibrcom 249	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) → (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)))
30	29	rexlimdva 3157	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) → (𝑓 ∈ 𝑇 ∧ 𝑡 ∈ 𝐸)))
31	30	pm4.71rd 569	. . . . . . 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 4592	. . . . . . . . 9 ⊢ (𝑡 ∈ {𝑂} ↔ 𝑡 = 𝑂)
33	32	anbi2i 631	. . . . . . . 8 ⊢ ((∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
34		r19.41v 3186	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂) ↔ (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
35	33, 34	bitr4i 280	. . . . . . 7 ⊢ ((∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
36		df-3an 1097	. . . . . . 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 316	. . . . . 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 281	. . . . 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 285	. . . 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 5814	. . 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 2791	. 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 2760	. . . . 5 ⊢ (𝑔 = ⟨𝑓, 𝑡⟩ → (𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ ⟨𝑓, 𝑡⟩ = ⟨(𝑠‘𝐹), 𝑂⟩))
43		vex 3452	. . . . . 6 ⊢ 𝑓 ∈ V
44		vex 3452	. . . . . 6 ⊢ 𝑡 ∈ V
45	43, 44	opth 5438	. . . . 5 ⊢ (⟨𝑓, 𝑡⟩ = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂))
46	42, 45	bitrdi 289	. . . 4 ⊢ (𝑔 = ⟨𝑓, 𝑡⟩ → (𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂)))
47	46	rexbidv 3180	. . 3 ⊢ (𝑔 = ⟨𝑓, 𝑡⟩ → (∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩ ↔ ∃𝑠 ∈ 𝐸 (𝑓 = (𝑠‘𝐹) ∧ 𝑡 = 𝑂)))
48	47	rabxp 5688	. 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 2809	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 𝑂⟩})

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1095   = wceq 1554   ∈ wcel 2136  ∃wrex 3080  {crab 3408  {csn 4576  ⟨cop 4582   class class class wbr 5094  {copab 5156   ↦ cmpt 5175   I cid 5534   × cxp 5638   ↾ cres 5642  ‘cfv 6510  Basecbs 17221  lecple 17269  HLchlt 39922  LHypclh 40556  LTrncltrn 40673  trLctrl 40730  TEndoctendo 41324  DIsoAcdia 41600  DIsoBcdib 41710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707  ax-riotaBAD 39525

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-iin 4946  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-1st 7959  df-2nd 7960  df-undef 8241  df-map 8798  df-proset 18302  df-poset 18321  df-plt 18336  df-lub 18352  df-glb 18353  df-join 18354  df-meet 18355  df-p0 18431  df-p1 18432  df-lat 18440  df-clat 18507  df-oposet 39748  df-ol 39750  df-oml 39751  df-covers 39838  df-ats 39839  df-atl 39870  df-cvlat 39894  df-hlat 39923  df-llines 40070  df-lplanes 40071  df-lvols 40072  df-lines 40073  df-psubsp 40075  df-pmap 40076  df-padd 40368  df-lhyp 40560  df-laut 40561  df-ldil 40676  df-ltrn 40677  df-trl 40731  df-tendo 41327  df-disoa 41601  df-dib 41711

This theorem is referenced by:  dib1dim2  41740