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Theorem dvhb1dimN 40160
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 8019 . . . . 5 (𝑔 ∈ (𝑇 Γ— 𝐸) β†’ (𝑔 = ⟨(π‘ β€˜πΉ), 0 ⟩ ↔ ((1st β€˜π‘”) = (π‘ β€˜πΉ) ∧ (2nd β€˜π‘”) = 0 )))
21adantl 480 . . . 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 3176 . . 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 3186 . . . 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 6903 . . . . . . . 8 (1st β€˜π‘”) ∈ V
6 eqeq1 2734 . . . . . . . . 9 (𝑓 = (1st β€˜π‘”) β†’ (𝑓 = (π‘ β€˜πΉ) ↔ (1st β€˜π‘”) = (π‘ β€˜πΉ)))
76rexbidv 3176 . . . . . . . 8 (𝑓 = (1st β€˜π‘”) β†’ (βˆƒπ‘  ∈ 𝐸 𝑓 = (π‘ β€˜πΉ) ↔ βˆƒπ‘  ∈ 𝐸 (1st β€˜π‘”) = (π‘ β€˜πΉ)))
85, 7elab 3667 . . . . . . 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 40159 . . . . . . . . 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 479 . . . . . . . 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 2817 . . . . . . 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 284 . . . . . 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 8009 . . . . . . . 8 (𝑔 ∈ (𝑇 Γ— 𝐸) β†’ (1st β€˜π‘”) ∈ 𝑇)
1918adantl 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ (1st β€˜π‘”) ∈ 𝑇)
20 fveq2 6890 . . . . . . . . 9 (𝑓 = (1st β€˜π‘”) β†’ (π‘…β€˜π‘“) = (π‘…β€˜(1st β€˜π‘”)))
2120breq1d 5157 . . . . . . . 8 (𝑓 = (1st β€˜π‘”) β†’ ((π‘…β€˜π‘“) ≀ (π‘…β€˜πΉ) ↔ (π‘…β€˜(1st β€˜π‘”)) ≀ (π‘…β€˜πΉ)))
2221elrab3 3683 . . . . . . 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 278 . . . . 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 628 . . . 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 282 . . 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 278 . 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 3437 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 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆƒwrex 3068  {crab 3430  βŸ¨cop 4633   class class class wbr 5147   ↦ cmpt 5230   I cid 5572   Γ— cxp 5673   β†Ύ cres 5677  β€˜cfv 6542  1st c1st 7975  2nd c2nd 7976  lecple 17208  HLchlt 38523  LHypclh 39158  LTrncltrn 39275  trLctrl 39332  TEndoctendo 39926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-riotaBAD 38126
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-undef 8260  df-map 8824  df-proset 18252  df-poset 18270  df-plt 18287  df-lub 18303  df-glb 18304  df-join 18305  df-meet 18306  df-p0 18382  df-p1 18383  df-lat 18389  df-clat 18456  df-oposet 38349  df-ol 38351  df-oml 38352  df-covers 38439  df-ats 38440  df-atl 38471  df-cvlat 38495  df-hlat 38524  df-llines 38672  df-lplanes 38673  df-lvols 38674  df-lines 38675  df-psubsp 38677  df-pmap 38678  df-padd 38970  df-lhyp 39162  df-laut 39163  df-ldil 39278  df-ltrn 39279  df-trl 39333  df-tendo 39929
This theorem is referenced by: (None)
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