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Theorem dib1dim 38293
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 = (𝑇 ↦ ( I ↾ 𝐵))
dib1dim.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dib1dim (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 𝑂⟩})
Distinct variable groups:   𝐵,   𝑔,𝑠,𝐸   𝑔,𝐹,𝑠   𝐻,𝑠   ,𝑠,𝐾   𝑔,𝑂,𝑠   𝑅,𝑠   𝑔,,𝑇,𝑠   ,𝑊,𝑠
Allowed substitution hints:   𝐵(𝑔,𝑠)   𝑅(𝑔,)   𝐸()   𝐹()   𝐻(𝑔,)   𝐼(𝑔,,𝑠)   𝐾(𝑔)   𝑂()   𝑊(𝑔)

Proof of Theorem dib1dim
Dummy variables 𝑓 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 485 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐾 ∈ HL ∧ 𝑊𝐻))
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)
62, 3, 4, 5trlcl 37292 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ 𝐵)
7 eqid 2819 . . . . 5 (le‘𝐾) = (le‘𝐾)
87, 3, 4, 5trlle 37312 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹)(le‘𝐾)𝑊)
9 dib1dim.o . . . . 5 𝑂 = (𝑇 ↦ ( I ↾ 𝐵))
10 eqid 2819 . . . . 5 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
11 dib1dim.i . . . . 5 𝐼 = ((DIsoB‘𝐾)‘𝑊)
122, 7, 3, 4, 9, 10, 11dibval2 38272 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑅𝐹) ∈ 𝐵 ∧ (𝑅𝐹)(le‘𝐾)𝑊)) → (𝐼‘(𝑅𝐹)) = ((((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) × {𝑂}))
131, 6, 8, 12syl12anc 834 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = ((((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) × {𝑂}))
14 relxp 5566 . . . 4 Rel ((((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) × {𝑂})
15 opelxp 5584 . . . . 5 (⟨𝑓, 𝑡⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) × {𝑂}) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) ∧ 𝑡 ∈ {𝑂}))
16 dib1dim.e . . . . . . . . 9 𝐸 = ((TEndo‘𝐾)‘𝑊)
173, 4, 5, 16, 10dia1dim 38189 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) = {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)})
1817abeq2d 2945 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) ↔ ∃𝑠𝐸 𝑓 = (𝑠𝐹)))
1918anbi1d 631 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) ∧ 𝑡 ∈ {𝑂}) ↔ (∃𝑠𝐸 𝑓 = (𝑠𝐹) ∧ 𝑡 ∈ {𝑂})))
203, 4, 16tendocl 37895 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → (𝑠𝐹) ∈ 𝑇)
21203expa 1113 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸) ∧ 𝐹𝑇) → (𝑠𝐹) ∈ 𝑇)
2221an32s 650 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → (𝑠𝐹) ∈ 𝑇)
232, 3, 4, 16, 9tendo0cl 37918 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂𝐸)
2423ad2antrr 724 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → 𝑂𝐸)
2522, 24jca 514 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → ((𝑠𝐹) ∈ 𝑇𝑂𝐸))
26 eleq1 2898 . . . . . . . . . . 11 (𝑓 = (𝑠𝐹) → (𝑓𝑇 ↔ (𝑠𝐹) ∈ 𝑇))
27 eleq1 2898 . . . . . . . . . . 11 (𝑡 = 𝑂 → (𝑡𝐸𝑂𝐸))
2826, 27bi2anan9 637 . . . . . . . . . 10 ((𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂) → ((𝑓𝑇𝑡𝐸) ↔ ((𝑠𝐹) ∈ 𝑇𝑂𝐸)))
2925, 28syl5ibrcom 249 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → ((𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂) → (𝑓𝑇𝑡𝐸)))
3029rexlimdva 3282 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂) → (𝑓𝑇𝑡𝐸)))
3130pm4.71rd 565 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂) ↔ ((𝑓𝑇𝑡𝐸) ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))))
32 velsn 4575 . . . . . . . . 9 (𝑡 ∈ {𝑂} ↔ 𝑡 = 𝑂)
3332anbi2i 624 . . . . . . . 8 ((∃𝑠𝐸 𝑓 = (𝑠𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ (∃𝑠𝐸 𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))
34 r19.41v 3345 . . . . . . . 8 (∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂) ↔ (∃𝑠𝐸 𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))
3533, 34bitr4i 280 . . . . . . 7 ((∃𝑠𝐸 𝑓 = (𝑠𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))
36 df-3an 1084 . . . . . . 7 ((𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂)) ↔ ((𝑓𝑇𝑡𝐸) ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂)))
3731, 35, 363bitr4g 316 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((∃𝑠𝐸 𝑓 = (𝑠𝐹) ∧ 𝑡 ∈ {𝑂}) ↔ (𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))))
3819, 37bitrd 281 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) ∧ 𝑡 ∈ {𝑂}) ↔ (𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))))
3915, 38syl5bb 285 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (⟨𝑓, 𝑡⟩ ∈ ((((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) × {𝑂}) ↔ (𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))))
4014, 39opabbi2dv 5713 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((((DIsoA‘𝐾)‘𝑊)‘(𝑅𝐹)) × {𝑂}) = {⟨𝑓, 𝑡⟩ ∣ (𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))})
4113, 40eqtrd 2854 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = {⟨𝑓, 𝑡⟩ ∣ (𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))})
42 eqeq1 2823 . . . . 5 (𝑔 = ⟨𝑓, 𝑡⟩ → (𝑔 = ⟨(𝑠𝐹), 𝑂⟩ ↔ ⟨𝑓, 𝑡⟩ = ⟨(𝑠𝐹), 𝑂⟩))
43 vex 3496 . . . . . 6 𝑓 ∈ V
44 vex 3496 . . . . . 6 𝑡 ∈ V
4543, 44opth 5359 . . . . 5 (⟨𝑓, 𝑡⟩ = ⟨(𝑠𝐹), 𝑂⟩ ↔ (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))
4642, 45syl6bb 289 . . . 4 (𝑔 = ⟨𝑓, 𝑡⟩ → (𝑔 = ⟨(𝑠𝐹), 𝑂⟩ ↔ (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂)))
4746rexbidv 3295 . . 3 (𝑔 = ⟨𝑓, 𝑡⟩ → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 𝑂⟩ ↔ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂)))
4847rabxp 5593 . 2 {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 𝑂⟩} = {⟨𝑓, 𝑡⟩ ∣ (𝑓𝑇𝑡𝐸 ∧ ∃𝑠𝐸 (𝑓 = (𝑠𝐹) ∧ 𝑡 = 𝑂))}
4941, 48syl6eqr 2872 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 𝑂⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1082   = wceq 1531  wcel 2108  wrex 3137  {crab 3140  {csn 4559  cop 4565   class class class wbr 5057  {copab 5119  cmpt 5137   I cid 5452   × cxp 5546  cres 5550  cfv 6348  Basecbs 16475  lecple 16564  HLchlt 36478  LHypclh 37112  LTrncltrn 37229  trLctrl 37286  TEndoctendo 37880  DIsoAcdia 38156  DIsoBcdib 38266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-fal 1544  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-undef 7931  df-map 8400  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-llines 36626  df-lplanes 36627  df-lvols 36628  df-lines 36629  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-lhyp 37116  df-laut 37117  df-ldil 37232  df-ltrn 37233  df-trl 37287  df-tendo 37883  df-disoa 38157  df-dib 38267
This theorem is referenced by:  dib1dim2  38296
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