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Theorem dvhb1dimN 36591
Description: Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dvhb1dim.l = (le‘𝐾)
dvhb1dim.h 𝐻 = (LHyp‘𝐾)
dvhb1dim.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhb1dim.r 𝑅 = ((trL‘𝐾)‘𝑊)
dvhb1dim.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhb1dim.o 0 = (𝑇 ↦ ( I ↾ 𝐵))
Assertion
Ref Expression
dvhb1dimN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩} = {𝑔 ∈ (𝑇 × 𝐸) ∣ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )})
Distinct variable groups:   ,𝑠   𝐸,𝑠   𝑔,𝑠,𝐹   𝑔,𝐻,𝑠   𝑔,𝐾,𝑠   0 ,𝑠   𝑅,𝑠   𝑔,,𝑇,𝑠   𝑔,𝑊,𝑠
Allowed substitution hints:   𝐵(𝑔,,𝑠)   𝑅(𝑔,)   𝐸(𝑔,)   𝐹()   𝐻()   𝐾()   (𝑔,)   𝑊()   0 (𝑔,)

Proof of Theorem dvhb1dimN
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 eqop 7252 . . . . 5 (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
21adantl 481 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
32rexbidv 3081 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
4 r19.41v 3118 . . . 4 (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ))
5 fvex 6239 . . . . . . . 8 (1st𝑔) ∈ V
6 eqeq1 2655 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑓 = (𝑠𝐹) ↔ (1st𝑔) = (𝑠𝐹)))
76rexbidv 3081 . . . . . . . 8 (𝑓 = (1st𝑔) → (∃𝑠𝐸 𝑓 = (𝑠𝐹) ↔ ∃𝑠𝐸 (1st𝑔) = (𝑠𝐹)))
85, 7elab 3382 . . . . . . 7 ((1st𝑔) ∈ {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} ↔ ∃𝑠𝐸 (1st𝑔) = (𝑠𝐹))
9 dvhb1dim.l . . . . . . . . . 10 = (le‘𝐾)
10 dvhb1dim.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
11 dvhb1dim.t . . . . . . . . . 10 𝑇 = ((LTrn‘𝐾)‘𝑊)
12 dvhb1dim.r . . . . . . . . . 10 𝑅 = ((trL‘𝐾)‘𝑊)
13 dvhb1dim.e . . . . . . . . . 10 𝐸 = ((TEndo‘𝐾)‘𝑊)
149, 10, 11, 12, 13dva1dim 36590 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1514adantr 480 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1615eleq2d 2716 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((1st𝑔) ∈ {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} ↔ (1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)}))
178, 16syl5bbr 274 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ↔ (1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)}))
18 xp1st 7242 . . . . . . . 8 (𝑔 ∈ (𝑇 × 𝐸) → (1st𝑔) ∈ 𝑇)
1918adantl 481 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st𝑔) ∈ 𝑇)
20 fveq2 6229 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑅𝑓) = (𝑅‘(1st𝑔)))
2120breq1d 4695 . . . . . . . 8 (𝑓 = (1st𝑔) → ((𝑅𝑓) (𝑅𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2221elrab3 3397 . . . . . . 7 ((1st𝑔) ∈ 𝑇 → ((1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)} ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2319, 22syl 17 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)} ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2417, 23bitrd 268 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2524anbi1d 741 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
264, 25syl5bb 272 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
273, 26bitrd 268 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
2827rabbidva 3219 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩} = {𝑔 ∈ (𝑇 × 𝐸) ∣ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  {crab 2945  cop 4216   class class class wbr 4685  cmpt 4762   I cid 5052   × cxp 5141  cres 5145  cfv 5926  1st c1st 7208  2nd c2nd 7209  lecple 15995  HLchlt 34955  LHypclh 35588  LTrncltrn 35705  trLctrl 35763  TEndoctendo 36357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-riotaBAD 34557
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-undef 7444  df-map 7901  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tendo 36360
This theorem is referenced by: (None)
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