Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dia0eldmN | Structured version Visualization version GIF version |
Description: The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dia0eldm.z | ⊢ 0 = (0.‘𝐾) |
dia0eldm.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dia0eldm.i | ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dia0eldmN | ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ dom 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlop 33667 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | 1 | adantr 480 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
3 | eqid 2610 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dia0eldm.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | op0cl 33489 | . . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾)) |
6 | 2, 5 | syl 17 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ (Base‘𝐾)) |
7 | dia0eldm.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
8 | 3, 7 | lhpbase 34302 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
9 | eqid 2610 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9, 4 | op0le 33491 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
11 | 1, 8, 10 | syl2an 493 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
12 | dia0eldm.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
13 | 3, 9, 7, 12 | diaeldm 35343 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( 0 ∈ dom 𝐼 ↔ ( 0 ∈ (Base‘𝐾) ∧ 0 (le‘𝐾)𝑊))) |
14 | 6, 11, 13 | mpbir2and 959 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ dom 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 Basecbs 15695 lecple 15775 0.cp0 16860 OPcops 33477 HLchlt 33655 LHypclh 34288 DIsoAcdia 35335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-glb 16798 df-p0 16862 df-oposet 33481 df-ol 33483 df-oml 33484 df-hlat 33656 df-lhyp 34292 df-disoa 35336 |
This theorem is referenced by: (None) |
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