Theorem diaelrnN 41674

Description: Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
diaelrn.h	⊢ 𝐻 = (LHyp‘𝐾)
diaelrn.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaelrn.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

Assertion

Ref	Expression
diaelrnN	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆 ⊆ 𝑇)

Proof of Theorem diaelrnN

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2764	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2764	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		diaelrn.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		diaelrn.i	. . . . 5 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
5	1, 2, 3, 4	diafn 41663	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6		fvelrnb 6929	. . . 4 ⊢ (𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → (𝑆 ∈ ran 𝐼 ↔ ∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼‘𝑥) = 𝑆))
7	5, 6	syl 17	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ ran 𝐼 ↔ ∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼‘𝑥) = 𝑆))
8		breq1 5105	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
9	8	elrab 3652	. . . . 5 ⊢ (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊))
10		diaelrn.t	. . . . . . . 8 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11	1, 2, 3, 10, 4	diass 41671	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼‘𝑥) ⊆ 𝑇)
12	11	ex 416	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼‘𝑥) ⊆ 𝑇))
13		sseq1 3963	. . . . . . 7 ⊢ ((𝐼‘𝑥) = 𝑆 → ((𝐼‘𝑥) ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
14	13	biimpcd 251	. . . . . 6 ⊢ ((𝐼‘𝑥) ⊆ 𝑇 → ((𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇))
15	12, 14	syl6 35	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇)))
16	9, 15	biimtrid 244	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → ((𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇)))
17	16	rexlimdv 3163	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇))
18	7, 17	sylbid 242	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ ran 𝐼 → 𝑆 ⊆ 𝑇))
19	18	imp 410	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆 ⊆ 𝑇)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  ∃wrex 3088  {crab 3416   ⊆ wss 3906   class class class wbr 5102  ran crn 5650   Fn wfn 6518  ‘cfv 6523  Basecbs 17247  lecple 17295  LHypclh 40613  LTrncltrn 40730  DIsoAcdia 41657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-disoa 41658

This theorem is referenced by:  dvadiaN  41757  djaclN  41765  djajN  41766