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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 765 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐾 ∈ 𝑉) | |
2 | ldil1o.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2821 | . . 3 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
4 | ldil1o.d | . . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ldillaut 37262 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾)) |
6 | ldil1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 6, 3 | laut1o 37236 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ (LAut‘𝐾)) → 𝐹:𝐵–1-1-onto→𝐵) |
8 | 1, 5, 7 | syl2anc 586 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹:𝐵–1-1-onto→𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 –1-1-onto→wf1o 6354 ‘cfv 6355 Basecbs 16483 LHypclh 37135 LAutclaut 37136 LDilcldil 37251 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408 df-laut 37140 df-ldil 37255 |
This theorem is referenced by: ldilcnv 37266 ldilco 37267 |
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