Theorem ldilval 40232

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 2733	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
5		ldilval.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	isldil 40229	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))
7		simpr 484	. . . 4 ⊢ ((𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)) → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))
8	6, 7	biimtrdi 253	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))
9		breq1 5096	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
10		fveq2 6828	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2749	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋))
13	9, 12	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) ↔ (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
14	13	rspccv 3570	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
15	14	impd 410	. . 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 1110	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5093  ‘cfv 6486  Basecbs 17122  lecple 17170  LHypclh 40103  LAutclaut 40104  LDilcldil 40219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ldil 40223

This theorem is referenced by:  ldilcnv  40234  ldilco  40235  ltrnval1  40253