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Mathbox for Norm Megill |
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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 3492 | . 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) | |
2 | polfval.p | . . 3 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
3 | fveq2 6902 | . . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾)) | |
4 | polfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | eqtr4di 2786 | . . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴) |
6 | 5 | pweqd 4623 | . . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴) |
7 | fveq2 6902 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = (pmap‘𝐾)) | |
8 | polfval.m | . . . . . . . . . 10 ⊢ 𝑀 = (pmap‘𝐾) | |
9 | 7, 8 | eqtr4di 2786 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = 𝑀) |
10 | fveq2 6902 | . . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = (oc‘𝐾)) | |
11 | polfval.o | . . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾) | |
12 | 10, 11 | eqtr4di 2786 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = ⊥ ) |
13 | 12 | fveq1d 6904 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → ((oc‘ℎ)‘𝑝) = ( ⊥ ‘𝑝)) |
14 | 9, 13 | fveq12d 6909 | . . . . . . . 8 ⊢ (ℎ = 𝐾 → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝))) |
15 | 14 | adantr 479 | . . . . . . 7 ⊢ ((ℎ = 𝐾 ∧ 𝑝 ∈ 𝑚) → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝))) |
16 | 15 | iineq2dv 5025 | . . . . . 6 ⊢ (ℎ = 𝐾 → ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) |
17 | 5, 16 | ineq12d 4215 | . . . . 5 ⊢ (ℎ = 𝐾 → ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) |
18 | 6, 17 | mpteq12dv 5243 | . . . 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 39408 | . . . 4 ⊢ ⊥𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))))) | |
20 | 4 | fvexi 6916 | . . . . . 6 ⊢ 𝐴 ∈ V |
21 | 20 | pwex 5384 | . . . . 5 ⊢ 𝒫 𝐴 ∈ V |
22 | 21 | mptex 7241 | . . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) ∈ V |
23 | 18, 19, 22 | fvmpt 7010 | . . 3 ⊢ (𝐾 ∈ V → (⊥𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
24 | 2, 23 | eqtrid 2780 | . 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 1533 ∈ wcel 2098 Vcvv 3473 ∩ cin 3948 𝒫 cpw 4606 ∩ ciin 5001 ↦ cmpt 5235 ‘cfv 6553 occoc 17248 Atomscatm 38767 pmapcpmap 39002 ⊥𝑃cpolN 39407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-polarityN 39408 |
This theorem is referenced by: polvalN 39410 |
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