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Theorem dib1dim 40036
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 𝐡 = (Baseβ€˜πΎ)
dib1dim.h 𝐻 = (LHypβ€˜πΎ)
dib1dim.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dib1dim.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dib1dim.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dib1dim.o 𝑂 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
dib1dim.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dib1dim (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (πΌβ€˜(π‘…β€˜πΉ)) = {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), π‘‚βŸ©})
Distinct variable groups:   𝐡,β„Ž   𝑔,𝑠,𝐸   𝑔,𝐹,𝑠   𝐻,𝑠   β„Ž,𝑠,𝐾   𝑔,𝑂,𝑠   𝑅,𝑠   𝑔,β„Ž,𝑇,𝑠   β„Ž,π‘Š,𝑠
Allowed substitution hints:   𝐡(𝑔,𝑠)   𝑅(𝑔,β„Ž)   𝐸(β„Ž)   𝐹(β„Ž)   𝐻(𝑔,β„Ž)   𝐼(𝑔,β„Ž,𝑠)   𝐾(𝑔)   𝑂(β„Ž)   π‘Š(𝑔)

Proof of Theorem dib1dim
Dummy variables 𝑓 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 dib1dim.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
3 dib1dim.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
4 dib1dim.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
5 dib1dim.r . . . . 5 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
62, 3, 4, 5trlcl 39035 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) ∈ 𝐡)
7 eqid 2733 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
87, 3, 4, 5trlle 39055 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ)(leβ€˜πΎ)π‘Š)
9 dib1dim.o . . . . 5 𝑂 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
10 eqid 2733 . . . . 5 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
11 dib1dim.i . . . . 5 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
122, 7, 3, 4, 9, 10, 11dibval2 40015 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((π‘…β€˜πΉ) ∈ 𝐡 ∧ (π‘…β€˜πΉ)(leβ€˜πΎ)π‘Š)) β†’ (πΌβ€˜(π‘…β€˜πΉ)) = ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) Γ— {𝑂}))
131, 6, 8, 12syl12anc 836 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (πΌβ€˜(π‘…β€˜πΉ)) = ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) Γ— {𝑂}))
14 relxp 5695 . . . 4 Rel ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) Γ— {𝑂})
15 opelxp 5713 . . . . 5 (βŸ¨π‘“, π‘‘βŸ© ∈ ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) Γ— {𝑂}) ↔ (𝑓 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) ∧ 𝑑 ∈ {𝑂}))
16 dib1dim.e . . . . . . . . 9 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
173, 4, 5, 16, 10dia1dim 39932 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) = {𝑓 ∣ βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ)})
1817eqabrd 2877 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝑓 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) ↔ βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ)))
1918anbi1d 631 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((𝑓 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) ∧ 𝑑 ∈ {𝑂}) ↔ (βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 ∈ {𝑂})))
203, 4, 16tendocl 39638 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘ β€˜πΉ) ∈ 𝑇)
21203expa 1119 . . . . . . . . . . . 12 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) ∧ 𝐹 ∈ 𝑇) β†’ (π‘ β€˜πΉ) ∈ 𝑇)
2221an32s 651 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) β†’ (π‘ β€˜πΉ) ∈ 𝑇)
232, 3, 4, 16, 9tendo0cl 39661 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑂 ∈ 𝐸)
2423ad2antrr 725 . . . . . . . . . . 11 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) β†’ 𝑂 ∈ 𝐸)
2522, 24jca 513 . . . . . . . . . 10 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) β†’ ((π‘ β€˜πΉ) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸))
26 eleq1 2822 . . . . . . . . . . 11 (𝑓 = (π‘ β€˜πΉ) β†’ (𝑓 ∈ 𝑇 ↔ (π‘ β€˜πΉ) ∈ 𝑇))
27 eleq1 2822 . . . . . . . . . . 11 (𝑑 = 𝑂 β†’ (𝑑 ∈ 𝐸 ↔ 𝑂 ∈ 𝐸))
2826, 27bi2anan9 638 . . . . . . . . . 10 ((𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂) β†’ ((𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸) ↔ ((π‘ β€˜πΉ) ∈ 𝑇 ∧ 𝑂 ∈ 𝐸)))
2925, 28syl5ibrcom 246 . . . . . . . . 9 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) β†’ ((𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂) β†’ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸)))
3029rexlimdva 3156 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂) β†’ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸)))
3130pm4.71rd 564 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂) ↔ ((𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸) ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))))
32 velsn 4645 . . . . . . . . 9 (𝑑 ∈ {𝑂} ↔ 𝑑 = 𝑂)
3332anbi2i 624 . . . . . . . 8 ((βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 ∈ {𝑂}) ↔ (βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))
34 r19.41v 3189 . . . . . . . 8 (βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂) ↔ (βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))
3533, 34bitr4i 278 . . . . . . 7 ((βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 ∈ {𝑂}) ↔ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))
36 df-3an 1090 . . . . . . 7 ((𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂)) ↔ ((𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸) ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂)))
3731, 35, 363bitr4g 314 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 ∈ {𝑂}) ↔ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))))
3819, 37bitrd 279 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((𝑓 ∈ (((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) ∧ 𝑑 ∈ {𝑂}) ↔ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))))
3915, 38bitrid 283 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βŸ¨π‘“, π‘‘βŸ© ∈ ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) Γ— {𝑂}) ↔ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))))
4014, 39opabbi2dv 5850 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((((DIsoAβ€˜πΎ)β€˜π‘Š)β€˜(π‘…β€˜πΉ)) Γ— {𝑂}) = {βŸ¨π‘“, π‘‘βŸ© ∣ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))})
4113, 40eqtrd 2773 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (πΌβ€˜(π‘…β€˜πΉ)) = {βŸ¨π‘“, π‘‘βŸ© ∣ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))})
42 eqeq1 2737 . . . . 5 (𝑔 = βŸ¨π‘“, π‘‘βŸ© β†’ (𝑔 = ⟨(π‘ β€˜πΉ), π‘‚βŸ© ↔ βŸ¨π‘“, π‘‘βŸ© = ⟨(π‘ β€˜πΉ), π‘‚βŸ©))
43 vex 3479 . . . . . 6 𝑓 ∈ V
44 vex 3479 . . . . . 6 𝑑 ∈ V
4543, 44opth 5477 . . . . 5 (βŸ¨π‘“, π‘‘βŸ© = ⟨(π‘ β€˜πΉ), π‘‚βŸ© ↔ (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))
4642, 45bitrdi 287 . . . 4 (𝑔 = βŸ¨π‘“, π‘‘βŸ© β†’ (𝑔 = ⟨(π‘ β€˜πΉ), π‘‚βŸ© ↔ (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂)))
4746rexbidv 3179 . . 3 (𝑔 = βŸ¨π‘“, π‘‘βŸ© β†’ (βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), π‘‚βŸ© ↔ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂)))
4847rabxp 5725 . 2 {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), π‘‚βŸ©} = {βŸ¨π‘“, π‘‘βŸ© ∣ (𝑓 ∈ 𝑇 ∧ 𝑑 ∈ 𝐸 ∧ βˆƒπ‘  ∈ 𝐸 (𝑓 = (π‘ β€˜πΉ) ∧ 𝑑 = 𝑂))}
4941, 48eqtr4di 2791 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (πΌβ€˜(π‘…β€˜πΉ)) = {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), π‘‚βŸ©})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  {crab 3433  {csn 4629  βŸ¨cop 4635   class class class wbr 5149  {copab 5211   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675   β†Ύ cres 5679  β€˜cfv 6544  Basecbs 17144  lecple 17204  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  trLctrl 39029  TEndoctendo 39623  DIsoAcdia 39899  DIsoBcdib 40009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-undef 8258  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030  df-tendo 39626  df-disoa 39900  df-dib 40010
This theorem is referenced by:  dib1dim2  40039
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