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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2pol0N | Structured version Visualization version GIF version |
Description: The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2pol0.o | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
2pol0N | ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | 2pol0.o | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
3 | 1, 2 | pol0N 37044 | . . 3 ⊢ (𝐾 ∈ HL → ( ⊥ ‘∅) = (Atoms‘𝐾)) |
4 | 3 | fveq2d 6673 | . 2 ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ( ⊥ ‘(Atoms‘𝐾))) |
5 | 1, 2 | pol1N 37045 | . 2 ⊢ (𝐾 ∈ HL → ( ⊥ ‘(Atoms‘𝐾)) = ∅) |
6 | 4, 5 | eqtrd 2856 | 1 ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∅c0 4290 ‘cfv 6354 Atomscatm 36398 HLchlt 36485 ⊥𝑃cpolN 37037 |
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 ax-riotaBAD 36088 |
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-rmo 3146 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-iin 4921 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-undef 7938 df-proset 17537 df-poset 17555 df-plt 17567 df-lub 17583 df-glb 17584 df-join 17585 df-meet 17586 df-p0 17648 df-p1 17649 df-lat 17655 df-clat 17717 df-oposet 36311 df-ol 36313 df-oml 36314 df-covers 36401 df-ats 36402 df-atl 36433 df-cvlat 36457 df-hlat 36486 df-pmap 36639 df-polarityN 37038 |
This theorem is referenced by: pcl0N 37057 0psubclN 37078 |
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