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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldil1o | Structured version Visualization version GIF version |
Description: A lattice dilation is a one-to-one onto function. (Contributed by NM, 19-Apr-2013.) |
Ref | Expression |
---|---|
ldil1o.b | ⊢ 𝐵 = (Base‘𝐾) |
ldil1o.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ldil1o.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ldil1o | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐵–1-1-onto→𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 767 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐾 ∈ 𝑉) | |
2 | ldil1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2734 | . . 3 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
4 | ldil1o.d | . . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ldillaut 40093 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾)) |
6 | ldil1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 6, 3 | laut1o 40067 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾)) → 𝐹:𝐵–1-1-onto→𝐵) |
8 | 1, 5, 7 | syl2anc 584 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 –1-1-onto→wf1o 6561 ‘cfv 6562 Basecbs 17244 LHypclh 39966 LAutclaut 39967 LDilcldil 40082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-laut 39971 df-ldil 40086 |
This theorem is referenced by: ldilcnv 40097 ldilco 40098 |
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