Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 3924 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
2 | eqid 2610 | . . . 4 ⊢ (oc‘𝐾) = (oc‘𝐾) | |
3 | polssat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | eqid 2610 | . . . 4 ⊢ (pmap‘𝐾) = (pmap‘𝐾) | |
5 | polssat.p | . . . 4 ⊢ ⊥ = (⊥𝑃‘𝐾) | |
6 | 2, 3, 4, 5 | polvalN 34209 | . . 3 ⊢ ((𝐾 ∈ 𝐵 ∧ ∅ ⊆ 𝐴) → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
7 | 1, 6 | mpan2 703 | . 2 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)))) |
8 | 0iin 4514 | . . . 4 ⊢ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝)) = V | |
9 | 8 | ineq2i 3773 | . . 3 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = (𝐴 ∩ V) |
10 | inv1 3922 | . . 3 ⊢ (𝐴 ∩ V) = 𝐴 | |
11 | 9, 10 | eqtri 2632 | . 2 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ ∅ ((pmap‘𝐾)‘((oc‘𝐾)‘𝑝))) = 𝐴 |
12 | 7, 11 | syl6eq 2660 | 1 ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∩ ciin 4456 ‘cfv 5804 occoc 15776 Atomscatm 33568 pmapcpmap 33801 ⊥𝑃cpolN 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-polarityN 34207 |
This theorem is referenced by: 2pol0N 34215 1psubclN 34248 osumcllem9N 34268 pexmidN 34273 pexmidlem6N 34279 pexmidALTN 34282 |
Copyright terms: Public domain | W3C validator |