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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 36513 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐾 ∈ OP) |
3 | eqid 2821 | . . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dia0eldm.z | . . . 4 ⊢ 0 = (0.‘𝐾) | |
5 | 3, 4 | op0cl 36335 | . . 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 37149 | . . 3 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
9 | eqid 2821 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
10 | 3, 9, 4 | op0le 36337 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑊 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑊) |
11 | 1, 8, 10 | syl2an 597 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 (le‘𝐾)𝑊) |
12 | dia0eldm.i | . . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) | |
13 | 3, 9, 7, 12 | diaeldm 38187 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( 0 ∈ dom 𝐼 ↔ ( 0 ∈ (Base‘𝐾) ∧ 0 (le‘𝐾)𝑊))) |
14 | 6, 11, 13 | mpbir2and 711 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 ∈ dom 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 Basecbs 16483 lecple 16572 0.cp0 17647 OPcops 36323 HLchlt 36501 LHypclh 37135 DIsoAcdia 38179 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
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 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-glb 17585 df-p0 17649 df-oposet 36327 df-ol 36329 df-oml 36330 df-hlat 36502 df-lhyp 37139 df-disoa 38180 |
This theorem is referenced by: (None) |
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