Theorem dvhb1dimN 41846

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 8031	. . . . 5 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
2	1	adantl 487	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
3	2	rexbidv 3188	. . 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 3194	. . . 4 ⊢ (∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ))
5		fvex 6895	. . . . . . . 8 ⊢ (1st ‘𝑔) ∈ V
6		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑓 = (𝑠‘𝐹) ↔ (1st ‘𝑔) = (𝑠‘𝐹)))
7	6	rexbidv 3188	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ↔ ∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹)))
8	5, 7	elab 3636	. . . . . . 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 41845	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)})
15	14	adantr 486	. . . . . . . 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 2848	. . . . . . 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 8021	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (1st ‘𝑔) ∈ 𝑇)
19	18	adantl 487	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st ‘𝑔) ∈ 𝑇)
20		fveq2 6882	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑅‘𝑓) = (𝑅‘(1st ‘𝑔)))
21	20	breq1d 5117	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → ((𝑅‘𝑓) ≤ (𝑅‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
22	21	elrab3 3649	. . . . . . 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 643	. . . 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 )})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088  {crab 3414  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   I cid 5553   × cxp 5657   ↾ cres 5661  ‘cfv 6537  1^st c1st 7987  2^nd c2nd 7988  lecple 17353  HLchlt 40210  LHypclh 40844  LTrncltrn 40961  trLctrl 41018  TEndoctendo 41612

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-riotaBAD 39813

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-undef 8274  df-map 8831  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-p1 18516  df-lat 18524  df-clat 18591  df-oposet 40036  df-ol 40038  df-oml 40039  df-covers 40126  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211  df-llines 40358  df-lplanes 40359  df-lvols 40360  df-lines 40361  df-psubsp 40363  df-pmap 40364  df-padd 40656  df-lhyp 40848  df-laut 40849  df-ldil 40964  df-ltrn 40965  df-trl 41019  df-tendo 41615

This theorem is used by: (None)