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Mirrors > Home > MPE Home > Th. List > Mathboxes > ldillaut | Structured version Visualization version GIF version |
Description: A lattice dilation is an automorphism. (Contributed by NM, 20-May-2012.) |
Ref | Expression |
---|---|
ldillaut.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ldillaut.i | ⊢ 𝐼 = (LAut‘𝐾) |
ldillaut.d | ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ldillaut | ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | eqid 2823 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | ldillaut.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | ldillaut.i | . . 3 ⊢ 𝐼 = (LAut‘𝐾) | |
5 | ldillaut.d | . . 3 ⊢ 𝐷 = ((LDil‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isldil 37248 | . 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝐹 ∈ 𝐷 ↔ (𝐹 ∈ 𝐼 ∧ ∀𝑥 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑊 → (𝐹‘𝑥) = 𝑥)))) |
7 | 6 | simprbda 501 | 1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝐷) → 𝐹 ∈ 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 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-pr 5332 |
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-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-ldil 37242 |
This theorem is referenced by: ldil1o 37250 ldilcnv 37253 ldilco 37254 ltrnlaut 37261 |
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