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| Mirrors > Home > MPE Home > Th. List > Mathboxes > polfvalN | Structured version Visualization version GIF version | ||
| Description: 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 |
|---|---|
| polfvalN | ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) | |
| 2 | polfval.p | . . 3 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
| 3 | fveq2 6906 | . . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾)) | |
| 4 | polfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴) |
| 6 | 5 | pweqd 4617 | . . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴) |
| 7 | fveq2 6906 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = (pmap‘𝐾)) | |
| 8 | polfval.m | . . . . . . . . . 10 ⊢ 𝑀 = (pmap‘𝐾) | |
| 9 | 7, 8 | eqtr4di 2795 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = 𝑀) |
| 10 | fveq2 6906 | . . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = (oc‘𝐾)) | |
| 11 | polfval.o | . . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾) | |
| 12 | 10, 11 | eqtr4di 2795 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = ⊥ ) |
| 13 | 12 | fveq1d 6908 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → ((oc‘ℎ)‘𝑝) = ( ⊥ ‘𝑝)) |
| 14 | 9, 13 | fveq12d 6913 | . . . . . . . 8 ⊢ (ℎ = 𝐾 → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝))) |
| 15 | 14 | adantr 480 | . . . . . . 7 ⊢ ((ℎ = 𝐾 ∧ 𝑝 ∈ 𝑚) → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝))) |
| 16 | 15 | iineq2dv 5017 | . . . . . 6 ⊢ (ℎ = 𝐾 → ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) |
| 17 | 5, 16 | ineq12d 4221 | . . . . 5 ⊢ (ℎ = 𝐾 → ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) |
| 18 | 6, 17 | mpteq12dv 5233 | . . . 4 ⊢ (ℎ = 𝐾 → (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)))) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
| 19 | df-polarityN 39905 | . . . 4 ⊢ ⊥𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))))) | |
| 20 | 4 | fvexi 6920 | . . . . . 6 ⊢ 𝐴 ∈ V |
| 21 | 20 | pwex 5380 | . . . . 5 ⊢ 𝒫 𝐴 ∈ V |
| 22 | 21 | mptex 7243 | . . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) ∈ V |
| 23 | 18, 19, 22 | fvmpt 7016 | . . 3 ⊢ (𝐾 ∈ V → (⊥𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
| 24 | 2, 23 | eqtrid 2789 | . 2 ⊢ (𝐾 ∈ V → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
| 25 | 1, 24 | syl 17 | 1 ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 𝒫 cpw 4600 ∩ ciin 4992 ↦ cmpt 5225 ‘cfv 6561 occoc 17305 Atomscatm 39264 pmapcpmap 39499 ⊥𝑃cpolN 39904 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-polarityN 39905 |
| This theorem is referenced by: polvalN 39907 |
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