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| Mirrors > Home > MPE Home > Th. List > Mathboxes > polatN | Structured version Visualization version GIF version | ||
| Description: The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| polat.o | ⊢ ⊥ = (oc‘𝐾) | 
| polat.a | ⊢ 𝐴 = (Atoms‘𝐾) | 
| polat.m | ⊢ 𝑀 = (pmap‘𝐾) | 
| polat.p | ⊢ 𝑃 = (⊥𝑃‘𝐾) | 
| Ref | Expression | 
|---|---|
| polatN | ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | snssi 4808 | . . 3 ⊢ (𝑄 ∈ 𝐴 → {𝑄} ⊆ 𝐴) | |
| 2 | polat.o | . . . 4 ⊢ ⊥ = (oc‘𝐾) | |
| 3 | polat.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
| 4 | polat.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
| 5 | polat.p | . . . 4 ⊢ 𝑃 = (⊥𝑃‘𝐾) | |
| 6 | 2, 3, 4, 5 | polvalN 39907 | . . 3 ⊢ ((𝐾 ∈ OL ∧ {𝑄} ⊆ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)))) | 
| 7 | 1, 6 | sylan2 593 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)))) | 
| 8 | 2fveq3 6911 | . . . . 5 ⊢ (𝑝 = 𝑄 → (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) | |
| 9 | 8 | iinxsng 5088 | . . . 4 ⊢ (𝑄 ∈ 𝐴 → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) | 
| 10 | 9 | adantl 481 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝)) = (𝑀‘( ⊥ ‘𝑄))) | 
| 11 | 10 | ineq2d 4220 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ ∩ 𝑝 ∈ {𝑄} (𝑀‘( ⊥ ‘𝑝))) = (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄)))) | 
| 12 | olop 39215 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
| 13 | eqid 2737 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
| 14 | 13, 3 | atbase 39290 | . . . . 5 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) | 
| 15 | 13, 2 | opoccl 39195 | . . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑄 ∈ (Base‘𝐾)) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) | 
| 16 | 12, 14, 15 | syl2an 596 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) | 
| 17 | 13, 3, 4 | pmapssat 39761 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ ( ⊥ ‘𝑄) ∈ (Base‘𝐾)) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴) | 
| 18 | 16, 17 | syldan 591 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴) | 
| 19 | sseqin2 4223 | . . 3 ⊢ ((𝑀‘( ⊥ ‘𝑄)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄))) | |
| 20 | 18, 19 | sylib 218 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝐴 ∩ (𝑀‘( ⊥ ‘𝑄))) = (𝑀‘( ⊥ ‘𝑄))) | 
| 21 | 7, 11, 20 | 3eqtrd 2781 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 {csn 4626 ∩ ciin 4992 ‘cfv 6561 Basecbs 17247 occoc 17305 OPcops 39173 OLcol 39175 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-ov 7434 df-oposet 39177 df-ol 39179 df-ats 39268 df-pmap 39506 df-polarityN 39905 | 
| This theorem is referenced by: 2polatN 39934 | 
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