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Mirrors > Home > MPE Home > Th. List > Mathboxes > polvalN | Structured version Visualization version GIF version |
Description: Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
polfval.o | ⊢ ⊥ = (oc‘𝐾) |
polfval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
polfval.m | ⊢ 𝑀 = (pmap‘𝐾) |
polfval.p | ⊢ 𝑃 = (⊥𝑃‘𝐾) |
Ref | Expression |
---|---|
polvalN | ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | polfval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | 1 | fvexi 6907 | . . 3 ⊢ 𝐴 ∈ V |
3 | 2 | elpw2 5344 | . 2 ⊢ (𝑋 ∈ 𝒫 𝐴 ↔ 𝑋 ⊆ 𝐴) |
4 | polfval.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
5 | polfval.m | . . . . 5 ⊢ 𝑀 = (pmap‘𝐾) | |
6 | polfval.p | . . . . 5 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
7 | 4, 1, 5, 6 | polfvalN 39616 | . . . 4 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
8 | 7 | fveq1d 6895 | . . 3 ⊢ (𝐾 ∈ 𝐵 → (𝑃‘𝑋) = ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋)) |
9 | iineq1 5010 | . . . . 5 ⊢ (𝑚 = 𝑋 → ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)) = ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))) | |
10 | 9 | ineq2d 4210 | . . . 4 ⊢ (𝑚 = 𝑋 → (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
11 | eqid 2726 | . . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) | |
12 | 2 | inex1 5314 | . . . 4 ⊢ (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝))) ∈ V |
13 | 10, 11, 12 | fvmpt 7001 | . . 3 ⊢ (𝑋 ∈ 𝒫 𝐴 → ((𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
14 | 8, 13 | sylan9eq 2786 | . 2 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝒫 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
15 | 3, 14 | sylan2br 593 | 1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∩ cin 3945 ⊆ wss 3946 𝒫 cpw 4597 ∩ ciin 4994 ↦ cmpt 5228 ‘cfv 6546 occoc 17269 Atomscatm 38974 pmapcpmap 39209 ⊥𝑃cpolN 39614 |
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 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-iin 4996 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-polarityN 39615 |
This theorem is referenced by: polval2N 39618 pol0N 39621 polcon3N 39629 polatN 39643 |
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