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 38931 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
8 | 7 | 3adant2r 1180 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
9 | 8 | fveq2d 6678 | . 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 38932 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑇)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥)) |
13 | 9, 12 | eqtrd 2773 | 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 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∃wrex 3054 {crab 3057 ⊆ wss 3843 ∅c0 4211 ∩ ciin 4882 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 lecple 16675 glbcglb 17669 meetcmee 17671 HLchlt 36987 LHypclh 37621 DIsoBcdib 38775 DIsoHcdih 38865 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-map 8439 df-proset 17654 df-poset 17672 df-plt 17684 df-lub 17700 df-glb 17701 df-join 17702 df-meet 17703 df-p0 17765 df-p1 17766 df-lat 17772 df-clat 17834 df-oposet 36813 df-ol 36815 df-oml 36816 df-covers 36903 df-ats 36904 df-atl 36935 df-cvlat 36959 df-hlat 36988 df-lhyp 37625 df-laut 37626 df-ldil 37741 df-ltrn 37742 df-trl 37796 df-disoa 38666 df-dib 38776 df-dih 38866 |
This theorem is referenced by: (None) |
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