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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 4005 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqid 2651 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2651 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polvalN 35509 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴) → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
7 | 1, 6 | mpan2 707 | . 2 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
8 | 0iin 4610 | . . . 4 ⊢ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V | |
9 | 8 | ineq2i 3844 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V) |
10 | inv1 4003 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
11 | 9, 10 | eqtri 2673 | . 2 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴 |
12 | 7, 11 | syl6eq 2701 | 1 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∩ cin 3606 ⊆ wss 3607 ∅c0 3948 ∩ ciin 4553 ‘cfv 5926 occoc 15996 Atomscatm 34868 pmapcpmap 35101 ⊥𝑃cpolN 35506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-polarityN 35507 |
This theorem is referenced by: 2pol0N 35515 1psubclN 35548 osumcllem9N 35568 pexmidN 35573 pexmidlem6N 35579 pexmidALTN 35582 |
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