Theorem pol0N 35979

Description: The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
polssat.a	⊢ 𝐴 = (Atoms‘𝐾)
polssat.p	⊢ ⊥ = (⊥𝑃‘𝐾)

Assertion

Ref	Expression
pol0N	⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴)

Proof of Theorem pol0N

Dummy variable 𝑝 is distinct from all other variables.

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)

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