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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 36378 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
3 | eqid 2818 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dia0eldm.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | op0cl 36200 | . . 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 37014 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
9 | eqid 2818 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9, 4 | op0le 36202 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
11 | 1, 8, 10 | syl2an 595 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
12 | dia0eldm.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
13 | 3, 9, 7, 12 | diaeldm 38052 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( 0 ∈ dom 𝐼 ↔ ( 0 ∈ (Base‘𝐾) ∧ 0 (le‘𝐾)𝑊))) |
14 | 6, 11, 13 | mpbir2and 709 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ dom 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 dom cdm 5548 ‘cfv 6348 Basecbs 16471 lecple 16560 0.cp0 17635 OPcops 36188 HLchlt 36366 LHypclh 37000 DIsoAcdia 38044 |
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 |
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-glb 17573 df-p0 17637 df-oposet 36192 df-ol 36194 df-oml 36195 df-hlat 36367 df-lhyp 37004 df-disoa 38045 |
This theorem is referenced by: (None) |
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