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Theorem dvhb1dimN 39857
Description: Two expressions for the 1-dimensional subspaces of vector space 𝐻, in the isomorphism B case where the second vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvhb1dim.l ≀ = (leβ€˜πΎ)
dvhb1dim.h 𝐻 = (LHypβ€˜πΎ)
dvhb1dim.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhb1dim.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dvhb1dim.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhb1dim.o 0 = (β„Ž ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
Assertion
Ref Expression
dvhb1dimN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩} = {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ ((π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )})
Distinct variable groups:   ≀ ,𝑠   𝐸,𝑠   𝑔,𝑠,𝐹   𝑔,𝐻,𝑠   𝑔,𝐾,𝑠   0 ,𝑠   𝑅,𝑠   𝑔,β„Ž,𝑇,𝑠   𝑔,π‘Š,𝑠
Allowed substitution hints:   𝐡(𝑔,β„Ž,𝑠)   𝑅(𝑔,β„Ž)   𝐸(𝑔,β„Ž)   𝐹(β„Ž)   𝐻(β„Ž)   𝐾(β„Ž)   ≀ (𝑔,β„Ž)   π‘Š(β„Ž)   0 (𝑔,β„Ž)
Proof of Theorem dvhb1dimN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqop 8017 . . . . 5 (𝑔 ∈ (𝑇 Γ— 𝐸) β†’ (𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩ ↔ ((1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
21adantl 483 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩ ↔ ((1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
32rexbidv 3179 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩ ↔ βˆƒπ‘  ∈ 𝐸 ((1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
4 r19.41v 3189 . . . 4 (βˆƒπ‘  ∈ 𝐸 ((1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 ) ↔ (βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 ))
5 fvex 6905 . . . . . . . 8 (1st β€˜π‘”) ∈ V
6 eqeq1 2737 . . . . . . . . 9 (𝑓 = (1st β€˜π‘”) β†’ (𝑓 = (π‘ β€˜πΉ) ↔ (1st β€˜π‘”) = (π‘ β€˜πΉ)))
76rexbidv 3179 . . . . . . . 8 (𝑓 = (1st β€˜π‘”) β†’ (βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ↔ βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ)))
85, 7elab 3669 . . . . . . 7 ((1st β€˜π‘”) ∈ {𝑓 ∣ βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ)} ↔ βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ))
9 dvhb1dim.l . . . . . . . . . 10 ≀ = (leβ€˜πΎ)
10 dvhb1dim.h . . . . . . . . . 10 𝐻 = (LHypβ€˜πΎ)
11 dvhb1dim.t . . . . . . . . . 10 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
12 dvhb1dim.r . . . . . . . . . 10 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
13 dvhb1dim.e . . . . . . . . . 10 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
149, 10, 11, 12, 13dva1dim 39856 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑓 ∣ βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ)} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ)})
1514adantr 482 . . . . . . . 8 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ {𝑓 ∣ βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ)} = {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ)})
1615eleq2d 2820 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((1st β€˜π‘”) ∈ {𝑓 ∣ βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ)} ↔ (1st β€˜π‘”) ∈ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ)}))
178, 16bitr3id 285 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ) ↔ (1st β€˜π‘”) ∈ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ)}))
18 xp1st 8007 . . . . . . . 8 (𝑔 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘”) ∈ 𝑇)
1918adantl 483 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘”) ∈ 𝑇)
20 fveq2 6892 . . . . . . . . 9 (𝑓 = (1st β€˜π‘”) β†’ (π‘…β€˜π‘“) = (π‘…β€˜(1st β€˜π‘”)))
2120breq1d 5159 . . . . . . . 8 (𝑓 = (1st β€˜π‘”) β†’ ((π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ) ↔ (π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ)))
2221elrab3 3685 . . . . . . 7 ((1st β€˜π‘”) ∈ 𝑇 β†’ ((1st β€˜π‘”) ∈ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ)} ↔ (π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ)))
2319, 22syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((1st β€˜π‘”) ∈ {𝑓 ∈ 𝑇 ∣ (π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ)} ↔ (π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ)))
2417, 23bitrd 279 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ) ↔ (π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ)))
2524anbi1d 631 . . . 4 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 ) ↔ ((π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
264, 25bitrid 283 . . 3 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (βˆƒπ‘  ∈ 𝐸 ((1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 ) ↔ ((π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
273, 26bitrd 279 . 2 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩ ↔ ((π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
2827rabbidva 3440 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩} = {𝑔 ∈ (𝑇 Γ— 𝐸) ∣ ((π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  {crab 3433  βŸ¨cop 4635   class class class wbr 5149   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675   β†Ύ cres 5679  β€˜cfv 6544  1st c1st 7973  2nd c2nd 7974  lecple 17204  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  trLctrl 39029  TEndoctendo 39623
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
This theorem is referenced by: (None)
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