Mathbox for Norm Megill |
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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 38432 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
8 | 7 | 3adant2r 1175 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
9 | 8 | fveq2d 6676 | . 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 38433 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑇)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥)) |
13 | 9, 12 | eqtrd 2858 | 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 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 {crab 3144 ⊆ wss 3938 ∅c0 4293 ∩ ciin 4922 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 lecple 16574 glbcglb 17555 meetcmee 17557 HLchlt 36488 LHypclh 37122 DIsoBcdib 38276 DIsoHcdih 38366 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-disoa 38167 df-dib 38277 df-dih 38367 |
This theorem is referenced by: (None) |
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