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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 3185 | . 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V) | |
2 | polfval.p | . . 3 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
3 | fveq2 6103 | . . . . . . 7 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = (Atoms‘𝐾)) | |
4 | polfval.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 3, 4 | syl6eqr 2662 | . . . . . 6 ⊢ (ℎ = 𝐾 → (Atoms‘ℎ) = 𝐴) |
6 | 5 | pweqd 4113 | . . . . 5 ⊢ (ℎ = 𝐾 → 𝒫 (Atoms‘ℎ) = 𝒫 𝐴) |
7 | fveq2 6103 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = (pmap‘𝐾)) | |
8 | polfval.m | . . . . . . . . . 10 ⊢ 𝑀 = (pmap‘𝐾) | |
9 | 7, 8 | syl6eqr 2662 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → (pmap‘ℎ) = 𝑀) |
10 | fveq2 6103 | . . . . . . . . . . 11 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = (oc‘𝐾)) | |
11 | polfval.o | . . . . . . . . . . 11 ⊢ ⊥ = (oc‘𝐾) | |
12 | 10, 11 | syl6eqr 2662 | . . . . . . . . . 10 ⊢ (ℎ = 𝐾 → (oc‘ℎ) = ⊥ ) |
13 | 12 | fveq1d 6105 | . . . . . . . . 9 ⊢ (ℎ = 𝐾 → ((oc‘ℎ)‘𝑝) = ( ⊥ ‘𝑝)) |
14 | 9, 13 | fveq12d 6109 | . . . . . . . 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 4479 | . . . . . 6 ⊢ (ℎ = 𝐾 → ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝)) = ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))) |
17 | 5, 16 | ineq12d 3777 | . . . . 5 ⊢ (ℎ = 𝐾 → ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) |
18 | 6, 17 | mpteq12dv 4663 | . . . 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 34207 | . . . 4 ⊢ ⊥𝑃 = (ℎ ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘ℎ) ↦ ((Atoms‘ℎ) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘ℎ)‘((oc‘ℎ)‘𝑝))))) | |
20 | fvex 6113 | . . . . . . 7 ⊢ (Atoms‘𝐾) ∈ V | |
21 | 4, 20 | eqeltri 2684 | . . . . . 6 ⊢ 𝐴 ∈ V |
22 | 21 | pwex 4774 | . . . . 5 ⊢ 𝒫 𝐴 ∈ V |
23 | 22 | mptex 6390 | . . . 4 ⊢ (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝)))) ∈ V |
24 | 18, 19, 23 | fvmpt 6191 | . . 3 ⊢ (𝐾 ∈ V → (⊥𝑃‘𝐾) = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) |
25 | 2, 24 | syl5eq 2656 | . 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 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 𝒫 cpw 4108 ∩ ciin 4456 ↦ cmpt 4643 ‘cfv 5804 occoc 15776 Atomscatm 33568 pmapcpmap 33801 ⊥𝑃cpolN 34206 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-polarityN 34207 |
This theorem is referenced by: polvalN 34209 |
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