Theorem dvhb1dimN 41706

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 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
dvhb1dimN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 0 ⟩} = {𝑔 ∈ (𝑇 × 𝐸) ∣ ((𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹) ∧ (2nd ‘𝑔) = 0 )})

Distinct variable groups:   ≤ ,𝑠   𝐸,𝑠   𝑔,𝑠,𝐹   𝑔,𝐻,𝑠   𝑔,𝐾,𝑠   0 ,𝑠   𝑅,𝑠   𝑔,ℎ,𝑇,𝑠   𝑔,𝑊,𝑠

Allowed substitution hints:   𝐵(𝑔,ℎ,𝑠)   𝑅(𝑔,ℎ)   𝐸(𝑔,ℎ)   𝐹(ℎ)   𝐻(ℎ)   𝐾(ℎ)   ≤ (𝑔,ℎ)   𝑊(ℎ)   0 (𝑔,ℎ)

Proof of Theorem dvhb1dimN

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqop 8027	. . . . 5 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
2	1	adantl 486	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
3	2	rexbidv 3187	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
4		r19.41v 3193	. . . 4 ⊢ (∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ))
5		fvex 6894	. . . . . . . 8 ⊢ (1st ‘𝑔) ∈ V
6		eqeq1 2765	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑓 = (𝑠‘𝐹) ↔ (1st ‘𝑔) = (𝑠‘𝐹)))
7	6	rexbidv 3187	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ↔ ∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹)))
8	5, 7	elab 3637	. . . . . . 7 ⊢ ((1st ‘𝑔) ∈ {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} ↔ ∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹))
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)
14	9, 10, 11, 12, 13	dva1dim 41705	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)})
15	14	adantr 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)})
16	15	eleq2d 2847	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((1st ‘𝑔) ∈ {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} ↔ (1st ‘𝑔) ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)}))
17	8, 16	bitr3id 288	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ↔ (1st ‘𝑔) ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)}))
18		xp1st 8017	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (1st ‘𝑔) ∈ 𝑇)
19	18	adantl 486	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st ‘𝑔) ∈ 𝑇)
20		fveq2 6881	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑅‘𝑓) = (𝑅‘(1st ‘𝑔)))
21	20	breq1d 5118	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → ((𝑅‘𝑓) ≤ (𝑅‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
22	21	elrab3 3650	. . . . . . 7 ⊢ ((1st ‘𝑔) ∈ 𝑇 → ((1st ‘𝑔) ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)} ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
23	19, 22	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((1st ‘𝑔) ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)} ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
24	17, 23	bitrd 282	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
25	24	anbi1d 642	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ ((𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
26	4, 25	bitrid 286	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ ((𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
27	3, 26	bitrd 282	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
28	27	rabbidva 3420	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠 ∈ 𝐸 𝑔 = ⟨(𝑠‘𝐹), 0 ⟩} = {𝑔 ∈ (𝑇 × 𝐸) ∣ ((𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹) ∧ (2nd ‘𝑔) = 0 )})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2141  {cab 2739  ∃wrex 3087  {crab 3414  ⟨cop 4594   class class class wbr 5108   ↦ cmpt 5191   I cid 5555   × cxp 5659   ↾ cres 5663  ‘cfv 6536  1^st c1st 7983  2^nd c2nd 7984  lecple 17316  HLchlt 40070  LHypclh 40704  LTrncltrn 40821  trLctrl 40878  TEndoctendo 41472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-riotaBAD 39673

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-undef 8268  df-map 8825  df-proset 18349  df-poset 18368  df-plt 18383  df-lub 18399  df-glb 18400  df-join 18401  df-meet 18402  df-p0 18478  df-p1 18479  df-lat 18487  df-clat 18554  df-oposet 39896  df-ol 39898  df-oml 39899  df-covers 39986  df-ats 39987  df-atl 40018  df-cvlat 40042  df-hlat 40071  df-llines 40218  df-lplanes 40219  df-lvols 40220  df-lines 40221  df-psubsp 40223  df-pmap 40224  df-padd 40516  df-lhyp 40708  df-laut 40709  df-ldil 40824  df-ltrn 40825  df-trl 40879  df-tendo 41475

This theorem is referenced by: (None)