Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhb1dimN Structured version   Visualization version   GIF version

Theorem dvhb1dimN 39205
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 7918 . . . . 5 (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
21adantl 482 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
32rexbidv 3172 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
4 r19.41v 3182 . . . 4 (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ))
5 fvex 6824 . . . . . . . 8 (1st𝑔) ∈ V
6 eqeq1 2741 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑓 = (𝑠𝐹) ↔ (1st𝑔) = (𝑠𝐹)))
76rexbidv 3172 . . . . . . . 8 (𝑓 = (1st𝑔) → (∃𝑠𝐸 𝑓 = (𝑠𝐹) ↔ ∃𝑠𝐸 (1st𝑔) = (𝑠𝐹)))
85, 7elab 3619 . . . . . . 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 39204 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1514adantr 481 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1615eleq2d 2823 . . . . . . 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 7908 . . . . . . . 8 (𝑔 ∈ (𝑇 × 𝐸) → (1st𝑔) ∈ 𝑇)
1918adantl 482 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st𝑔) ∈ 𝑇)
20 fveq2 6811 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑅𝑓) = (𝑅‘(1st𝑔)))
2120breq1d 5097 . . . . . . . 8 (𝑓 = (1st𝑔) → ((𝑅𝑓) (𝑅𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2221elrab3 3635 . . . . . . 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 630 . . . 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 3411 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 396   = wceq 1540  wcel 2105  {cab 2714  wrex 3071  {crab 3404  cop 4577   class class class wbr 5087  cmpt 5170   I cid 5506   × cxp 5605  cres 5609  cfv 6465  1st c1st 7874  2nd c2nd 7875  lecple 17039  HLchlt 37568  LHypclh 38203  LTrncltrn 38320  trLctrl 38377  TEndoctendo 38971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7628  ax-riotaBAD 37171
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-id 5507  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-riota 7272  df-ov 7318  df-oprab 7319  df-mpo 7320  df-1st 7876  df-2nd 7877  df-undef 8136  df-map 8665  df-proset 18083  df-poset 18101  df-plt 18118  df-lub 18134  df-glb 18135  df-join 18136  df-meet 18137  df-p0 18213  df-p1 18214  df-lat 18220  df-clat 18287  df-oposet 37394  df-ol 37396  df-oml 37397  df-covers 37484  df-ats 37485  df-atl 37516  df-cvlat 37540  df-hlat 37569  df-llines 37717  df-lplanes 37718  df-lvols 37719  df-lines 37720  df-psubsp 37722  df-pmap 37723  df-padd 38015  df-lhyp 38207  df-laut 38208  df-ldil 38323  df-ltrn 38324  df-trl 38378  df-tendo 38974
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator