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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > pol0N | Structured version Visualization version GIF version |
Description: The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polssat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polssat.p | ⊢ ⊥ = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
pol0N | ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4397 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqid 2728 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2728 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polvalN 39378 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴) → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
7 | 1, 6 | mpan2 690 | . 2 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
8 | 0iin 5067 | . . . 4 ⊢ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V | |
9 | 8 | ineq2i 4209 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V) |
10 | inv1 4395 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
11 | 9, 10 | eqtri 2756 | . 2 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴 |
12 | 7, 11 | eqtrdi 2784 | 1 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ∅c0 4323 ∩ ciin 4997 ‘cfv 6548 occoc 17241 Atomscatm 38735 pmapcpmap 38970 ⊥𝑃cpolN 39375 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-polarityN 39376 |
This theorem is referenced by: 2pol0N 39384 1psubclN 39417 osumcllem9N 39437 pexmidN 39442 pexmidlem6N 39448 pexmidALTN 39451 |
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