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Mirrors > Home > MPE Home > Th. List > Mathboxes > pmapglb | Structured version Visualization version GIF version |
Description: The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.) |
Ref | Expression |
---|---|
pmapglb.b | ⊢ 𝐵 = (Base‘𝐾) |
pmapglb.g | ⊢ 𝐺 = (glb‘𝐾) |
pmapglb.m | ⊢ 𝑀 = (pmap‘𝐾) |
Ref | Expression |
---|---|
pmapglb | ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3144 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥)) | |
2 | equcom 2021 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | anbi1ci 627 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆)) |
4 | 3 | exbii 1844 | . . . . . . 7 ⊢ (∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆)) |
5 | eleq1w 2895 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆)) | |
6 | 5 | equsexvw 2007 | . . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆) ↔ 𝑦 ∈ 𝑆) |
7 | 1, 4, 6 | 3bitri 299 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ 𝑦 ∈ 𝑆) |
8 | 7 | abbii 2886 | . . . . 5 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥} = {𝑦 ∣ 𝑦 ∈ 𝑆} |
9 | abid2 2957 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑆} = 𝑆 | |
10 | 8, 9 | eqtr2i 2845 | . . . 4 ⊢ 𝑆 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥} |
11 | 10 | fveq2i 6667 | . . 3 ⊢ (𝐺‘𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}) |
12 | 11 | fveq2i 6667 | . 2 ⊢ (𝑀‘(𝐺‘𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) |
13 | dfss3 3955 | . . 3 ⊢ (𝑆 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ 𝐵) | |
14 | pmapglb.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
15 | pmapglb.g | . . . 4 ⊢ 𝐺 = (glb‘𝐾) | |
16 | pmapglb.m | . . . 4 ⊢ 𝑀 = (pmap‘𝐾) | |
17 | 14, 15, 16 | pmapglbx 36899 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
18 | 13, 17 | syl3an2b 1400 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
19 | 12, 18 | syl5eq 2868 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3935 ∅c0 4290 ∩ ciin 4912 ‘cfv 6349 Basecbs 16477 glbcglb 17547 HLchlt 36480 pmapcpmap 36627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-poset 17550 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-lat 17650 df-clat 17712 df-ats 36397 df-hlat 36481 df-pmap 36634 |
This theorem is referenced by: pmapglb2N 36901 pmapmeet 36903 |
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