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Mathbox for Norm Megill |
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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 3069 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ ∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥)) | |
2 | equcom 2015 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | anbi1ci 626 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆)) |
4 | 3 | exbii 1845 | . . . . . . 7 ⊢ (∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆)) |
5 | eleq1w 2822 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆)) | |
6 | 5 | equsexvw 2002 | . . . . . . 7 ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆) ↔ 𝑦 ∈ 𝑆) |
7 | 1, 4, 6 | 3bitri 297 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝑆 𝑦 = 𝑥 ↔ 𝑦 ∈ 𝑆) |
8 | 7 | abbii 2807 | . . . . 5 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥} = {𝑦 ∣ 𝑦 ∈ 𝑆} |
9 | abid2 2877 | . . . . 5 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑆} = 𝑆 | |
10 | 8, 9 | eqtr2i 2764 | . . . 4 ⊢ 𝑆 = {𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥} |
11 | 10 | fveq2i 6910 | . . 3 ⊢ (𝐺‘𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}) |
12 | 11 | fveq2i 6910 | . 2 ⊢ (𝑀‘(𝐺‘𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) |
13 | dfss3 3984 | . . 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 39752 | . . 3 ⊢ ((𝐾 ∈ HL ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
18 | 13, 17 | syl3an2b 1403 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
19 | 12, 18 | eqtrid 2787 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∅c0 4339 ∩ ciin 4997 ‘cfv 6563 Basecbs 17245 glbcglb 18368 HLchlt 39332 pmapcpmap 39480 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-poset 18371 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-lat 18490 df-clat 18557 df-ats 39249 df-hlat 39333 df-pmap 39487 |
This theorem is referenced by: pmapglb2N 39754 pmapmeet 39756 |
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