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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 3352 | . 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) | |
2 | polfval.p | . . 3 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
3 | fveq2 6353 | . . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾)) | |
4 | polfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | syl6eqr 2812 | . . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴) |
6 | 5 | pweqd 4307 | . . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴) |
7 | fveq2 6353 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = (pmap‘𝐾)) | |
8 | polfval.m | . . . . . . . . . 10 ⊢ 𝑀 = (pmap‘𝐾) | |
9 | 7, 8 | syl6eqr 2812 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = 𝑀) |
10 | fveq2 6353 | . . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = (oc‘𝐾)) | |
11 | polfval.o | . . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾) | |
12 | 10, 11 | syl6eqr 2812 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = ⊥ ) |
13 | 12 | fveq1d 6355 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → ((oc‘ℎ)‘𝑝) = ( ⊥ ‘𝑝)) |
14 | 9, 13 | fveq12d 6359 | . . . . . . . 8 ⊢ (ℎ = 𝐾 → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝))) |
15 | 14 | adantr 472 | . . . . . . 7 ⊢ ((ℎ = 𝐾 ∧ 𝑝 ∈ 𝑚) → ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = (𝑀‘( ⊥ ‘𝑝))) |
16 | 15 | iineq2dv 4695 | . . . . . 6 ⊢ (ℎ = 𝐾 → ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) |
17 | 5, 16 | ineq12d 3958 | . . . . 5 ⊢ (ℎ = 𝐾 → ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) |
18 | 6, 17 | mpteq12dv 4885 | . . . 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 35710 | . . . 4 ⊢ ⊥𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))))) | |
20 | fvex 6363 | . . . . . . 7 ⊢ (Atoms‘𝐾) ∈ V | |
21 | 4, 20 | eqeltri 2835 | . . . . . 6 ⊢ 𝐴 ∈ V |
22 | 21 | pwex 4997 | . . . . 5 ⊢ 𝒫 𝐴 ∈ V |
23 | 22 | mptex 6651 | . . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) ∈ V |
24 | 18, 19, 23 | fvmpt 6445 | . . 3 ⊢ (𝐾 ∈ V → (⊥𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
25 | 2, 24 | syl5eq 2806 | . 2 ⊢ (𝐾 ∈ V → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
26 | 1, 25 | 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 1632 ∈ wcel 2139 Vcvv 3340 ∩ cin 3714 𝒫 cpw 4302 ∩ ciin 4673 ↦ cmpt 4881 ‘cfv 6049 occoc 16171 Atomscatm 35071 pmapcpmap 35304 ⊥𝑃cpolN 35709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-polarityN 35710 |
This theorem is referenced by: polvalN 35712 |
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