Theorem dvhb1dimN 42235

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 8067	. . . . 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 3190	. . 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 3196	. . . 4 ⊢ (∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ))
5		fvex 6921	. . . . . . . 8 ⊢ (1st ‘𝑔) ∈ V
6		eqeq1 2767	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑓 = (𝑠‘𝐹) ↔ (1st ‘𝑔) = (𝑠‘𝐹)))
7	6	rexbidv 3190	. . . . . . . 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 42234	. . . . . . . . 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 2850	. . . . . . 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 8057	. . . . . . . 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 6908	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑅‘𝑓) = (𝑅‘(1st ‘𝑔)))
21	20	breq1d 5116	. . . . . . . 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 3422	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 2147  {cab 2741  ∃wrex 3090  {crab 3416  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   I cid 5549   × cxp 5653   ↾ cres 5657  ‘cfv 6558  1^st c1st 8023  2^nd c2nd 8024  lecple 17464  HLchlt 40599  LHypclh 41233  LTrncltrn 41350  trLctrl 41407  TEndoctendo 42001

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-riotaBAD 40202

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-undef 8318  df-map 8877  df-proset 18497  df-poset 18516  df-plt 18531  df-lub 18547  df-glb 18548  df-join 18549  df-meet 18550  df-p0 18626  df-p1 18627  df-lat 18635  df-clat 18702  df-oposet 40425  df-ol 40427  df-oml 40428  df-covers 40515  df-ats 40516  df-atl 40547  df-cvlat 40571  df-hlat 40600  df-llines 40747  df-lplanes 40748  df-lvols 40749  df-lines 40750  df-psubsp 40752  df-pmap 40753  df-padd 41045  df-lhyp 41237  df-laut 41238  df-ldil 41353  df-ltrn 41354  df-trl 41408  df-tendo 42004

This theorem is used by: (None)