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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldilval | Structured version Visualization version GIF version |
Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ldilval.b | ⊢ 𝐵 = (Base‘𝐾) |
ldilval.l | ⊢ ≤ = (le‘𝐾) |
ldilval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ldilval.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ldilval | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
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 2610 | . . . . 5 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
5 | ldilval.d | . . . . 5 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isldil 34414 | . . . 4 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)))) |
7 | simpr 476 | . . . 4 ⊢ ((𝐹 ∈ (LAut‘𝐾) ∧ ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)) → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥)) | |
8 | 6, 7 | syl6bi 242 | . . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥))) |
9 | breq1 4586 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑊 ↔ 𝑋 ≤ 𝑊)) | |
10 | fveq2 6103 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
11 | id 22 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
12 | 10, 11 | eqeq12d 2625 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = 𝑥 ↔ (𝐹‘𝑋) = 𝑋)) |
13 | 9, 12 | imbi12d 333 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) ↔ (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋))) |
14 | 13 | rspccv 3279 | . . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → (𝑋 ∈ 𝐵 → (𝑋 ≤ 𝑊 → (𝐹‘𝑋) = 𝑋))) |
15 | 14 | impd 446 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ≤ 𝑊 → (𝐹‘𝑥) = 𝑥) → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → (𝐹‘𝑋) = 𝑋)) |
16 | 8, 15 | syl6 34 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 → ((𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) → (𝐹‘𝑋) = 𝑋))) |
17 | 16 | 3imp 1249 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷 ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹‘𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 LHypclh 34288 LAutclaut 34289 LDilcldil 34404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ldil 34408 |
This theorem is referenced by: ldilcnv 34419 ldilco 34420 ltrnval1 34438 |
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