Theorem ldilval 40620

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 2741	. . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾)
5		ldilval.d	. . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	isldil 40617	. . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))))
7		simpr 486	. . . 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 5078	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊))
10		fveq2 6831	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
12	10, 11	eqeq12d 2757	. . . . . 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 3559	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋)))
15	14	impd 412	. . 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 1117	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055   class class class wbr 5075  ‘cfv 6489  Basecbs 17174  lecple 17222  LHypclh 40491  LAutclaut 40492  LDilcldil 40607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ldil 40611

This theorem is referenced by:  ldilcnv  40622  ldilco  40623  ltrnval1  40641