Theorem diaelrnN 42293

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 2763	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . . . 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 42282	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑦 ∈ (Base‘𝐾) ∣ 𝑦(le‘𝐾)𝑊})
6		fvelrnb 6965	. . . 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 5109	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦(le‘𝐾)𝑊 ↔ 𝑥(le‘𝐾)𝑊))
9	8	elrab 3648	. . . . 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 42290	. . . . . . 7 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)) → (𝐼‘𝑥) ⊆ 𝑇)
12	11	ex 418	. . . . . 6 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) → (𝐼‘𝑥) ⊆ 𝑇))
13		sseq1 3938	. . . . . . 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 3165	. . 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 412	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆 ⊆ 𝑇)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416   ⊆ wss 3900   class class class wbr 5106  ran crn 5656   Fn wfn 6550  ‘cfv 6555  Basecbs 17413  lecple 17461  LHypclh 41232  LTrncltrn 41349  DIsoAcdia 42276

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-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-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-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 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-disoa 42277

This theorem is used by:  dvadiaN  42376  djaclN  42384  djajN  42385