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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 4199 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqid 2825 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2825 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polvalN 35975 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴) → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
7 | 1, 6 | mpan2 682 | . 2 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
8 | 0iin 4800 | . . . 4 ⊢ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V | |
9 | 8 | ineq2i 4040 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V) |
10 | inv1 4197 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
11 | 9, 10 | eqtri 2849 | . 2 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴 |
12 | 7, 11 | syl6eq 2877 | 1 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ∩ cin 3797 ⊆ wss 3798 ∅c0 4146 ∩ ciin 4743 ‘cfv 6127 occoc 16320 Atomscatm 35333 pmapcpmap 35567 ⊥𝑃cpolN 35972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-polarityN 35973 |
This theorem is referenced by: 2pol0N 35981 1psubclN 36014 osumcllem9N 36034 pexmidN 36039 pexmidlem6N 36045 pexmidALTN 36048 |
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