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 38310 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
8 | 7 | 3adant2r 1171 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐺‘𝑆) = (𝐺‘𝑇)) |
9 | 8 | fveq2d 6667 | . 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 38311 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑇)) = ∩ 𝑥 ∈ 𝑇 (𝐼‘𝑥)) |
13 | 9, 12 | eqtrd 2853 | 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 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∃wrex 3136 {crab 3139 ⊆ wss 3933 ∅c0 4288 ∩ ciin 4911 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 lecple 16560 glbcglb 17541 meetcmee 17543 HLchlt 36366 LHypclh 37000 DIsoBcdib 38154 DIsoHcdih 38244 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-proset 17526 df-poset 17544 df-plt 17556 df-lub 17572 df-glb 17573 df-join 17574 df-meet 17575 df-p0 17637 df-p1 17638 df-lat 17644 df-clat 17706 df-oposet 36192 df-ol 36194 df-oml 36195 df-covers 36282 df-ats 36283 df-atl 36314 df-cvlat 36338 df-hlat 36367 df-lhyp 37004 df-laut 37005 df-ldil 37120 df-ltrn 37121 df-trl 37175 df-disoa 38045 df-dib 38155 df-dih 38245 |
This theorem is referenced by: (None) |
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