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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 37140 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
5 | f1ocnvdm 7032 | . . . 4 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ 𝑃 ∈ 𝐵) → (◡𝐹‘𝑃) ∈ 𝐵) | |
6 | 4, 5 | stoic3 1768 | . . 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 37153 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (◡𝐹‘𝑃) ∈ 𝐵) → ((◡𝐹‘𝑃) ∈ 𝐴 ↔ (𝐹‘(◡𝐹‘𝑃)) ∈ 𝐴)) |
9 | 6, 8 | syld3an3 1401 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → ((◡𝐹‘𝑃) ∈ 𝐴 ↔ (𝐹‘(◡𝐹‘𝑃)) ∈ 𝐴)) |
10 | f1ocnvfv2 7025 | . . . 4 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ 𝑃 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑃)) = 𝑃) | |
11 | 4, 10 | stoic3 1768 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑃)) = 𝑃) |
12 | 11 | eleq1d 2894 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐵) → ((𝐹‘(◡𝐹‘𝑃)) ∈ 𝐴 ↔ 𝑃 ∈ 𝐴)) |
13 | 9, 12 | bitr2d 281 | 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ◡ccnv 5547 –1-1-onto→wf1o 6347 ‘cfv 6348 Basecbs 16471 Atomscatm 36279 HLchlt 36366 LHypclh 37000 LTrncltrn 37117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-plt 17556 df-glb 17573 df-p0 17637 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-hlat 36367 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 |
This theorem is referenced by: ltrncnvat 37157 |
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