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Theorem dvhb1dimN 36794
 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 7376 . . . . 5 (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
21adantl 473 . . . 4 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
32rexbidv 3190 . . 3 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩ ↔ ∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 )))
4 r19.41v 3227 . . . 4 (∃𝑠𝐸 ((1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ) ↔ (∃𝑠𝐸 (1st𝑔) = (𝑠𝐹) ∧ (2nd𝑔) = 0 ))
5 fvex 6363 . . . . . . . 8 (1st𝑔) ∈ V
6 eqeq1 2764 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑓 = (𝑠𝐹) ↔ (1st𝑔) = (𝑠𝐹)))
76rexbidv 3190 . . . . . . . 8 (𝑓 = (1st𝑔) → (∃𝑠𝐸 𝑓 = (𝑠𝐹) ↔ ∃𝑠𝐸 (1st𝑔) = (𝑠𝐹)))
85, 7elab 3490 . . . . . . 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 36793 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1514adantr 472 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → {𝑓 ∣ ∃𝑠𝐸 𝑓 = (𝑠𝐹)} = {𝑓𝑇 ∣ (𝑅𝑓) (𝑅𝐹)})
1615eleq2d 2825 . . . . . . 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 7366 . . . . . . . 8 (𝑔 ∈ (𝑇 × 𝐸) → (1st𝑔) ∈ 𝑇)
1918adantl 473 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st𝑔) ∈ 𝑇)
20 fveq2 6353 . . . . . . . . 9 (𝑓 = (1st𝑔) → (𝑅𝑓) = (𝑅‘(1st𝑔)))
2120breq1d 4814 . . . . . . . 8 (𝑓 = (1st𝑔) → ((𝑅𝑓) (𝑅𝐹) ↔ (𝑅‘(1st𝑔)) (𝑅𝐹)))
2221elrab3 3505 . . . . . . 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 743 . . . 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 3328 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 1632   ∈ wcel 2139  {cab 2746  ∃wrex 3051  {crab 3054  ⟨cop 4327   class class class wbr 4804   ↦ cmpt 4881   I cid 5173   × cxp 5264   ↾ cres 5268  ‘cfv 6049  1st c1st 7332  2nd c2nd 7333  lecple 16170  HLchlt 35158  LHypclh 35791  LTrncltrn 35908  trLctrl 35966  TEndoctendo 36560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-riotaBAD 34760 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-undef 7569  df-map 8027  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-p1 17261  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-llines 35305  df-lplanes 35306  df-lvols 35307  df-lines 35308  df-psubsp 35310  df-pmap 35311  df-padd 35603  df-lhyp 35795  df-laut 35796  df-ldil 35911  df-ltrn 35912  df-trl 35967  df-tendo 36563 This theorem is referenced by: (None)
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