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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 2025 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦) | |
3 | 2 | anbi1ci 627 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ (𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆)) |
4 | 3 | exbii 1848 | . . . . . . 7 ⊢ (∃𝑥(𝑥 ∈ 𝑆 ∧ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝑥 ∈ 𝑆)) |
5 | eleq1w 2895 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑆 ↔ 𝑦 ∈ 𝑆)) | |
6 | 5 | equsexvw 2011 | . . . . . . 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 6673 | . . 3 ⊢ (𝐺‘𝑆) = (𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥}) |
12 | 11 | fveq2i 6673 | . 2 ⊢ (𝑀‘(𝐺‘𝑆)) = (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑥 ∈ 𝑆 𝑦 = 𝑥})) |
13 | dfss3 3956 | . . 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 36920 | . . 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 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 ∅c0 4291 ∩ ciin 4920 ‘cfv 6355 Basecbs 16483 glbcglb 17553 HLchlt 36501 pmapcpmap 36648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-poset 17556 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-lat 17656 df-clat 17718 df-ats 36418 df-hlat 36502 df-pmap 36655 |
This theorem is referenced by: pmapglb2N 36922 pmapmeet 36924 |
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