Theorem ldilval 40701

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 2761	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
5		ldilval.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	isldil 40698	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))
7		simpr 488	. . . 4 ⊢ ((𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)) → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))
8	6, 7	biimtrdi 255	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
9		breq1 5102	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
10		fveq2 6863	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋))
13	9, 12	imbi12d 346	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) ↔ (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
14	13	rspccv 3578	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
15	14	impd 414	. . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → (𝐹‘𝑋) = 𝑋))
16	8, 15	syl6 35	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → (𝐹‘𝑋) = 𝑋)))
17	16	3imp 1122	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   class class class wbr 5099  ‘cfv 6517  Basecbs 17228  lecple 17276  LHypclh 40572  LAutclaut 40573  LDilcldil 40688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ldil 40692

This theorem is referenced by:  ldilcnv  40703  ldilco  40704  ltrnval1  40722