Theorem ldilval 40928

Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.)

Hypotheses

Ref	Expression
ldilval.b	⊢ 𝐵 = (Base‘𝐾)
ldilval.l	⊢ ≤ = (le‘𝐾)
ldilval.h	⊢ 𝐻 = (LHyp‘𝐾)
ldilval.d	⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)

Assertion

Ref	Expression
ldilval	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Proof of Theorem ldilval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldilval.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		ldilval.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		ldilval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		eqid 2766	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
5		ldilval.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	isldil 40925	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))
7		simpr 490	. . . 4 ⊢ ((𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)) → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))
8	6, 7	biimtrdi 256	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
9		breq1 5117	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
10		fveq2 6888	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2782	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋))
13	9, 12	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) ↔ (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
14	13	rspccv 3581	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
15	14	impd 416	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → (𝐹‘𝑋) = 𝑋))
16	8, 15	syl6 36	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → (𝐹‘𝑋) = 𝑋)))
17	16	3imp 1128	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3082   class class class wbr 5114  ‘cfv 6543  Basecbs 17294  lecple 17342  LHypclh 40799  LAutclaut 40800  LDilcldil 40915

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ldil 40919

This theorem is used by:  ldilcnv  40930  ldilco  40931  ltrnval1  40949