Theorem dvhb1dimN 41548

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 7997	. . . . 5 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
2	1	adantl 484	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (𝑔 = ⟨(𝑠‘𝐹), 0 ⟩ ↔ ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 )))
3	2	rexbidv 3176	. . 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 3182	. . . 4 ⊢ (∃𝑠 ∈ 𝐸 ((1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ) ↔ (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ∧ (2nd ‘𝑔) = 0 ))
5		fvex 6865	. . . . . . . 8 ⊢ (1st ‘𝑔) ∈ V
6		eqeq1 2756	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑓 = (𝑠‘𝐹) ↔ (1st ‘𝑔) = (𝑠‘𝐹)))
7	6	rexbidv 3176	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → (∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹) ↔ ∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹)))
8	5, 7	elab 3629	. . . . . . 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 41547	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑓 ∣ ∃𝑠 ∈ 𝐸 𝑓 = (𝑠‘𝐹)} = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ (𝑅‘𝐹)})
15	14	adantr 483	. . . . . . . 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 2838	. . . . . . 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 287	. . . . . 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 7987	. . . . . . . 8 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (1st ‘𝑔) ∈ 𝑇)
19	18	adantl 484	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (1st ‘𝑔) ∈ 𝑇)
20		fveq2 6852	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑔) → (𝑅‘𝑓) = (𝑅‘(1st ‘𝑔)))
21	20	breq1d 5100	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑔) → ((𝑅‘𝑓) ≤ (𝑅‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
22	21	elrab3 3642	. . . . . . 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 281	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → (∃𝑠 ∈ 𝐸 (1st ‘𝑔) = (𝑠‘𝐹) ↔ (𝑅‘(1st ‘𝑔)) ≤ (𝑅‘𝐹)))
25	24	anbi1d 639	. . . 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 285	. . 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 281	. 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 3410	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 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {cab 2730  ∃wrex 3076  {crab 3404  ⟨cop 4578   class class class wbr 5090   ↦ cmpt 5171   I cid 5530   × cxp 5634   ↾ cres 5638  ‘cfv 6506  1^st c1st 7953  2^nd c2nd 7954  lecple 17265  HLchlt 39912  LHypclh 40546  LTrncltrn 40663  trLctrl 40720  TEndoctendo 41314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-rep 5217  ax-sep 5236  ax-nul 5246  ax-pow 5312  ax-pr 5380  ax-un 7703  ax-riotaBAD 39515

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rmo 3357  df-reu 3358  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-pw 4547  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-iun 4941  df-iin 4942  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-f 6510  df-f1 6511  df-fo 6512  df-f1o 6513  df-fv 6514  df-riota 7338  df-ov 7384  df-oprab 7385  df-mpo 7386  df-1st 7955  df-2nd 7956  df-undef 8237  df-map 8794  df-proset 18298  df-poset 18317  df-plt 18332  df-lub 18348  df-glb 18349  df-join 18350  df-meet 18351  df-p0 18427  df-p1 18428  df-lat 18436  df-clat 18503  df-oposet 39738  df-ol 39740  df-oml 39741  df-covers 39828  df-ats 39829  df-atl 39860  df-cvlat 39884  df-hlat 39913  df-llines 40060  df-lplanes 40061  df-lvols 40062  df-lines 40063  df-psubsp 40065  df-pmap 40066  df-padd 40358  df-lhyp 40550  df-laut 40551  df-ldil 40666  df-ltrn 40667  df-trl 40721  df-tendo 41317

This theorem is referenced by: (None)