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Theorem dvhb1dimN 36999
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 7441 . . . . 5 (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
21adantl 474 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
32rexbidv 3231 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
4 r19.41v 3268 . . . 4 (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ))
5 fvex 6422 . . . . . . . 8 (1st𝑔) ∈ V
6 eqeq1 2801 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑓 = (𝑠𝐹) ↔ (1st𝑔) = (𝑠𝐹)))
76rexbidv 3231 . . . . . . . 8 (𝑓 = (1st𝑔) → (∃𝑠𝐸 𝑓 = (𝑠𝐹) ↔ ∃𝑠𝐸 (1st𝑔) = (𝑠𝐹)))
85, 7elab 3540 . . . . . . 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 36998 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1514adantr 473 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1615eleq2d 2862 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((1st𝑔) ∈ {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} ↔ (1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)}))
178, 16syl5bbr 277 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ↔ (1st𝑔) ∈ {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)}))
18 xp1st 7431 . . . . . . . 8 (𝑔 ∈ (𝑇 × 𝐸) → (1st𝑔) ∈ 𝑇)
1918adantl 474 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st𝑔) ∈ 𝑇)
20 fveq2 6409 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑅𝑓) = (𝑅‘(1st𝑔)))
2120breq1d 4851 . . . . . . . 8 (𝑓 = (1st𝑔) → ((𝑅𝑓) (𝑅𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2221elrab3 3556 . . . . . . 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 271 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2524anbi1d 624 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
264, 25syl5bb 275 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
273, 26bitrd 271 . 2 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )))
2827rabbidva 3370 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 198  wa 385   = wceq 1653  wcel 2157  {cab 2783  wrex 3088  {crab 3091  cop 4372   class class class wbr 4841  cmpt 4920   I cid 5217   × cxp 5308  cres 5312  cfv 6099  1st c1st 7397  2nd c2nd 7398  lecple 16271  HLchlt 35363  LHypclh 35997  LTrncltrn 36114  trLctrl 36171  TEndoctendo 36765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-riotaBAD 34966
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-undef 7635  df-map 8095  df-proset 17240  df-poset 17258  df-plt 17270  df-lub 17286  df-glb 17287  df-join 17288  df-meet 17289  df-p0 17351  df-p1 17352  df-lat 17358  df-clat 17420  df-oposet 35189  df-ol 35191  df-oml 35192  df-covers 35279  df-ats 35280  df-atl 35311  df-cvlat 35335  df-hlat 35364  df-llines 35511  df-lplanes 35512  df-lvols 35513  df-lines 35514  df-psubsp 35516  df-pmap 35517  df-padd 35809  df-lhyp 36001  df-laut 36002  df-ldil 36117  df-ltrn 36118  df-trl 36172  df-tendo 36768
This theorem is referenced by: (None)
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