Nearby theorems
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Mathbox for Norm Megill |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > latm32 | Structured version Visualization version GIF version | ||
| Description: A rearrangement of lattice meet. (in12 4213 analog.) (Contributed by NM, 13-Nov-2012.) |
| Ref | Expression |
|---|---|
| olmass.b | β’ π΅ = (BaseβπΎ) |
| olmass.m | β’ β§ = (meetβπΎ) |
| Ref | Expression |
|---|---|
| latm32 | β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = ((π β§ π) β§ π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ollat 38587 | . . . . 5 β’ (πΎ β OL β πΎ β Lat) | |
| 2 | olmass.b | . . . . . 6 β’ π΅ = (BaseβπΎ) | |
| 3 | olmass.m | . . . . . 6 β’ β§ = (meetβπΎ) | |
| 4 | 2, 3 | latmcom 18424 | . . . . 5 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 5 | 1, 4 | syl3an1 1160 | . . . 4 β’ ((πΎ β OL β§ π β π΅ β§ π β π΅) β (π β§ π) = (π β§ π)) |
| 6 | 5 | 3adant3r1 1179 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ π) = (π β§ π)) |
| 7 | 6 | oveq2d 7418 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β (π β§ (π β§ π)) = (π β§ (π β§ π))) |
| 8 | 2, 3 | latmassOLD 38603 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = (π β§ (π β§ π))) |
| 9 | simpl 482 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β πΎ β OL) | |
| 10 | simpr1 1191 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 11 | simpr3 1193 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 12 | simpr2 1192 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β π β π΅) | |
| 13 | 2, 3 | latmassOLD 38603 | . . 3 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = (π β§ (π β§ π))) |
| 14 | 9, 10, 11, 12, 13 | syl13anc 1369 | . 2 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = (π β§ (π β§ π))) |
| 15 | 7, 8, 14 | 3eqtr4d 2774 | 1 β’ ((πΎ β OL β§ (π β π΅ β§ π β π΅ β§ π β π΅)) β ((π β§ π) β§ π) = ((π β§ π) β§ π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6534 (class class class)co 7402 Basecbs 17149 meetcmee 18273 Latclat 18392 OLcol 38548 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-proset 18256 df-poset 18274 df-lub 18307 df-glb 18308 df-join 18309 df-meet 18310 df-lat 18393 df-oposet 38550 df-ol 38552 |
| This theorem is referenced by: cdleme20d 39687 dia2dimlem3 40441 |
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