Theorem dvhb1dimN 36999

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 = (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 7441	. . . . 5 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
2	1	adantl 474	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
3	2	rexbidv 3231	. . 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 3268	. . . 4 ⊢ (∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ))
5		fvex 6422	. . . . . . . 8 ⊢ (1st ‘𝑔) ∈ V
6		eqeq1 2801	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑓 = (𝑠‘𝐹) ↔ (1st ‘𝑔) = (𝑠‘𝐹)))
7	6	rexbidv 3231	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ↔ ∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹)))
8	5, 7	elab 3540	. . . . . . 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 36998	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)})
15	14	adantr 473	. . . . . . . 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 2862	. . . . . . 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	syl5bbr 277	. . . . . 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 7431	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (1st ‘𝑔) ∈ 𝑇)
19	18	adantl 474	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st ‘𝑔) ∈ 𝑇)
20		fveq2 6409	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑅‘𝑓) = (𝑅‘(1st ‘𝑔)))
21	20	breq1d 4851	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → ((𝑅‘𝑓) ≤ (𝑅‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
22	21	elrab3 3556	. . . . . . 7 ⊢ ((1st ‘𝑔) ∈ 𝑇 → ((1st ‘𝑔) ∈ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)} ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
23	19, 22	syl 17	. . . . . 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 271	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
25	24	anbi1d 624	. . . 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	syl5bb 275	. . 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 271	. 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 3370	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 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  {cab 2783  ∃wrex 3088  {crab 3091  ⟨cop 4372   class class class wbr 4841   ↦ cmpt 4920   I cid 5217   × cxp 5308   ↾ cres 5312  ‘cfv 6099  1^st c1st 7397  2^nd c2nd 7398  lecple 16271  HLchlt 35363  LHypclh 35997  LTrncltrn 36114  trLctrl 36171  TEndoctendo 36765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-riotaBAD 34966

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-iin 4711  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-riota 6837  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-undef 7635  df-map 8095  df-proset 17240  df-poset 17258  df-plt 17270  df-lub 17286  df-glb 17287  df-join 17288  df-meet 17289  df-p0 17351  df-p1 17352  df-lat 17358  df-clat 17420  df-oposet 35189  df-ol 35191  df-oml 35192  df-covers 35279  df-ats 35280  df-atl 35311  df-cvlat 35335  df-hlat 35364  df-llines 35511  df-lplanes 35512  df-lvols 35513  df-lines 35514  df-psubsp 35516  df-pmap 35517  df-padd 35809  df-lhyp 36001  df-laut 36002  df-ldil 36117  df-ltrn 36118  df-trl 36172  df-tendo 36768

This theorem is referenced by: (None)