| Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lautle | Structured version Visualization version GIF version | ||
| Description: Less-than or equal property of a lattice automorphism. (Contributed by NM, 19-May-2012.) |
| Ref | Expression |
|---|---|
| lautset.b | ⊢ 𝐵 = (Base‘𝐾) |
| lautset.l | ⊢ ≤ = (le‘𝐾) |
| lautset.i | ⊢ 𝐼 = (LAut‘𝐾) |
| Ref | Expression |
|---|---|
| lautle | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lautset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | lautset.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
| 3 | lautset.i | . . . 4 ⊢ 𝐼 = (LAut‘𝐾) | |
| 4 | 1, 2, 3 | islaut 40085 | . . 3 ⊢ (𝐾 ∈ 𝑉 → (𝐹 ∈ 𝐼 ↔ (𝐹:𝐵–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))))) |
| 5 | 4 | simplbda 499 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦))) |
| 6 | breq1 5146 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦)) | |
| 7 | fveq2 6906 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
| 8 | 7 | breq1d 5153 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦))) |
| 9 | 6, 8 | bibi12d 345 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑦 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦)))) |
| 10 | breq2 5147 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌)) | |
| 11 | fveq2 6906 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌)) | |
| 12 | 11 | breq2d 5155 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
| 13 | 10, 12 | bibi12d 345 | . . 3 ⊢ (𝑦 = 𝑌 → ((𝑋 ≤ 𝑦 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑦)) ↔ (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))) |
| 14 | 9, 13 | rspc2v 3633 | . 2 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ↔ (𝐹‘𝑥) ≤ (𝐹‘𝑦)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌)))) |
| 15 | 5, 14 | mpan9 506 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝐹‘𝑋) ≤ (𝐹‘𝑌))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 –1-1-onto→wf1o 6560 ‘cfv 6561 Basecbs 17247 lecple 17304 LAutclaut 39987 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-laut 39991 |
| This theorem is referenced by: lautcnvle 40091 lautlt 40093 lautj 40095 lautm 40096 lauteq 40097 lautco 40099 ltrnle 40131 |
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