![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihglblem3aN | Structured version Visualization version GIF version |
Description: Isomorphism H of a lattice glb. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihglblem.b | ⊢ 𝐵 = (Base‘𝐾) |
dihglblem.l | ⊢ ≤ = (le‘𝐾) |
dihglblem.m | ⊢ ∧ = (meet‘𝐾) |
dihglblem.g | ⊢ 𝐺 = (glb‘𝐾) |
dihglblem.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihglblem.t | ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)} |
dihglblem.i | ⊢ 𝐽 = ((DIsoB‘𝐾)‘𝑊) |
dihglblem.ih | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihglblem3aN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihglblem.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihglblem.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | dihglblem.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
4 | dihglblem.g | . . . . 5 ⊢ 𝐺 = (glb‘𝐾) | |
5 | dihglblem.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | dihglblem.t | . . . . 5 ⊢ 𝑇 = {𝑢 ∈ 𝐵 ∣ ∃𝑣 ∈ 𝑆 𝑢 = (𝑣 ∧ 𝑊)} | |
7 | 1, 2, 3, 4, 5, 6 | dihglblem2N 41276 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
8 | 7 | 3adant2r 1178 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
9 | 8 | fveq2d 6910 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = (𝐼‘(𝐺‘𝑇))) |
10 | dihglblem.i | . . 3 ⊢ 𝐽 = ((DIsoB‘𝐾)‘𝑊) | |
11 | dihglblem.ih | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
12 | 1, 2, 3, 4, 5, 6, 10, 11 | dihglblem3N 41277 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑇)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥)) |
13 | 9, 12 | eqtrd 2774 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∃wrex 3067 {crab 3432 ⊆ wss 3962 ∅c0 4338 ∩ ciin 4996 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 lecple 17304 glbcglb 18367 meetcmee 18369 HLchlt 39331 LHypclh 39966 DIsoBcdib 41120 DIsoHcdih 41210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-map 8866 df-proset 18351 df-poset 18370 df-plt 18387 df-lub 18403 df-glb 18404 df-join 18405 df-meet 18406 df-p0 18482 df-p1 18483 df-lat 18489 df-clat 18556 df-oposet 39157 df-ol 39159 df-oml 39160 df-covers 39247 df-ats 39248 df-atl 39279 df-cvlat 39303 df-hlat 39332 df-lhyp 39970 df-laut 39971 df-ldil 40086 df-ltrn 40087 df-trl 40141 df-disoa 41011 df-dib 41121 df-dih 41211 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |