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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrncnvatb | Structured version Visualization version GIF version |
Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.) |
Ref | Expression |
---|---|
ltrnatb.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrnatb.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrnatb.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrnatb.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
ltrncnvatb | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ (◡𝐹‘𝑃) ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrnatb.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | ltrnatb.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | ltrnatb.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
4 | 1, 2, 3 | ltrn1o 37259 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
5 | f1ocnvdm 7040 | . . . 4 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ 𝑃 ∈ 𝐵) → (◡𝐹‘𝑃) ∈ 𝐵) | |
6 | 4, 5 | stoic3 1773 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → (◡𝐹‘𝑃) ∈ 𝐵) |
7 | ltrnatb.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | 1, 7, 2, 3 | ltrnatb 37272 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (◡𝐹‘𝑃) ∈ 𝐵) → ((◡𝐹‘𝑃) ∈ 𝐴 ↔ (𝐹‘(◡𝐹‘𝑃)) ∈ 𝐴)) |
9 | 6, 8 | syld3an3 1405 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → ((◡𝐹‘𝑃) ∈ 𝐴 ↔ (𝐹‘(◡𝐹‘𝑃)) ∈ 𝐴)) |
10 | f1ocnvfv2 7033 | . . . 4 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ 𝑃 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑃)) = 𝑃) | |
11 | 4, 10 | stoic3 1773 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑃)) = 𝑃) |
12 | 11 | eleq1d 2897 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → ((𝐹‘(◡𝐹‘𝑃)) ∈ 𝐴 ↔ 𝑃 ∈ 𝐴)) |
13 | 9, 12 | bitr2d 282 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → (𝑃 ∈ 𝐴 ↔ (◡𝐹‘𝑃) ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ◡ccnv 5553 –1-1-onto→wf1o 6353 ‘cfv 6354 Basecbs 16482 Atomscatm 36398 HLchlt 36485 LHypclh 37119 LTrncltrn 37236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-map 8407 df-plt 17567 df-glb 17584 df-p0 17648 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-hlat 36486 df-lhyp 37123 df-laut 37124 df-ldil 37239 df-ltrn 37240 |
This theorem is referenced by: ltrncnvat 37276 |
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