Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 2823 | . . 3 ⊢ (LAut‘𝐾) = (LAut‘𝐾) | |
4 | ldil1o.d | . . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | ldillaut 37249 | . 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ (LAut‘𝐾)) |
6 | ldil1o.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | 6, 3 | laut1o 37223 | . 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 6356 ‘cfv 6357 Basecbs 16485 LHypclh 37122 LAutclaut 37123 LDilcldil 37238 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-laut 37127 df-ldil 37242 |
This theorem is referenced by: ldilcnv 37253 ldilco 37254 |
Copyright terms: Public domain | W3C validator |