Theorem diaelrnN 41704

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 2769	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2769	. . . . 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 41693	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6		fvelrnb 6939	. . . 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 18	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ ran 𝐼 ↔ ∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼‘𝑥) = 𝑆))
8		breq1 5113	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
9	8	elrab 3659	. . . . 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 41701	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼‘𝑥) ⊆ 𝑇)
12	11	ex 417	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼‘𝑥) ⊆ 𝑇))
13		sseq1 3970	. . . . . . 7 ⊢ ((𝐼‘𝑥) = 𝑆 → ((𝐼‘𝑥) ⊆ 𝑇 ↔ 𝑆 ⊆ 𝑇))
14	13	biimpcd 252	. . . . . 6 ⊢ ((𝐼‘𝑥) ⊆ 𝑇 → ((𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇))
15	12, 14	syl6 36	. . . . 5 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → ((𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇)))
16	9, 15	biimtrid 245	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} → ((𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇)))
17	16	rexlimdv 3170	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (∃𝑥 ∈ {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊} (𝐼‘𝑥) = 𝑆 → 𝑆 ⊆ 𝑇))
18	7, 17	sylbid 243	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ ran 𝐼 → 𝑆 ⊆ 𝑇))
19	18	imp 411	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆 ⊆ 𝑇)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  {crab 3423   ⊆ wss 3913   class class class wbr 5110  ran crn 5660   Fn wfn 6528  ‘cfv 6533  Basecbs 17265  lecple 17313  LHypclh 40643  LTrncltrn 40760  DIsoAcdia 41687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  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 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-disoa 41688

This theorem is referenced by:  dvadiaN  41787  djaclN  41795  djajN  41796