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Theorem dvhb1dimN 40685
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 = (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 8045 . . . . 5 (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
21adantl 480 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
32rexbidv 3169 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
4 r19.41v 3179 . . . 4 (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ))
5 fvex 6914 . . . . . . . 8 (1st𝑔) ∈ V
6 eqeq1 2730 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑓 = (𝑠𝐹) ↔ (1st𝑔) = (𝑠𝐹)))
76rexbidv 3169 . . . . . . . 8 (𝑓 = (1st𝑔) → (∃𝑠𝐸 𝑓 = (𝑠𝐹) ↔ ∃𝑠𝐸 (1st𝑔) = (𝑠𝐹)))
85, 7elab 3666 . . . . . . 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 40684 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1514adantr 479 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1615eleq2d 2812 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((1st𝑔) ∈ {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} ↔ (1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)}))
178, 16bitr3id 284 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ↔ (1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)}))
18 xp1st 8035 . . . . . . . 8 (𝑔 ∈ (𝑇 × 𝐸) → (1st𝑔) ∈ 𝑇)
1918adantl 480 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st𝑔) ∈ 𝑇)
20 fveq2 6901 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑅𝑓) = (𝑅‘(1st𝑔)))
2120breq1d 5163 . . . . . . . 8 (𝑓 = (1st𝑔) → ((𝑅𝑓) (𝑅𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2221elrab3 3682 . . . . . . 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 278 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2524anbi1d 629 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
264, 25bitrid 282 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
273, 26bitrd 278 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
2827rabbidva 3426 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 205  wa 394   = wceq 1534  wcel 2099  {cab 2703  wrex 3060  {crab 3419  cop 4639   class class class wbr 5153  cmpt 5236   I cid 5579   × cxp 5680  cres 5684  cfv 6554  1st c1st 8001  2nd c2nd 8002  lecple 17273  HLchlt 39048  LHypclh 39683  LTrncltrn 39800  trLctrl 39857  TEndoctendo 40451
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 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-riotaBAD 38651
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-iin 5004  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-undef 8288  df-map 8857  df-proset 18320  df-poset 18338  df-plt 18355  df-lub 18371  df-glb 18372  df-join 18373  df-meet 18374  df-p0 18450  df-p1 18451  df-lat 18457  df-clat 18524  df-oposet 38874  df-ol 38876  df-oml 38877  df-covers 38964  df-ats 38965  df-atl 38996  df-cvlat 39020  df-hlat 39049  df-llines 39197  df-lplanes 39198  df-lvols 39199  df-lines 39200  df-psubsp 39202  df-pmap 39203  df-padd 39495  df-lhyp 39687  df-laut 39688  df-ldil 39803  df-ltrn 39804  df-trl 39858  df-tendo 40454
This theorem is referenced by: (None)
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