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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlateq | Structured version Visualization version GIF version |
Description: The equality of two Hilbert lattice elements is determined by the atoms under them. (chrelat4i 32131 analog.) (Contributed by NM, 24-May-2012.) |
Ref | Expression |
---|---|
hlatle.b | β’ π΅ = (BaseβπΎ) |
hlatle.l | β’ β€ = (leβπΎ) |
hlatle.a | β’ π΄ = (AtomsβπΎ) |
Ref | Expression |
---|---|
hlateq | β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (βπ β π΄ (π β€ π β π β€ π) β π = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbiim 3107 | . . 3 β’ (βπ β π΄ (π β€ π β π β€ π) β (βπ β π΄ (π β€ π β π β€ π) β§ βπ β π΄ (π β€ π β π β€ π))) | |
2 | hlatle.b | . . . . 5 β’ π΅ = (BaseβπΎ) | |
3 | hlatle.l | . . . . 5 β’ β€ = (leβπΎ) | |
4 | hlatle.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
5 | 2, 3, 4 | hlatle 38780 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (π β€ π β βπ β π΄ (π β€ π β π β€ π))) |
6 | 2, 3, 4 | hlatle 38780 | . . . . 5 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (π β€ π β βπ β π΄ (π β€ π β π β€ π))) |
7 | 6 | 3com23 1123 | . . . 4 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (π β€ π β βπ β π΄ (π β€ π β π β€ π))) |
8 | 5, 7 | anbi12d 630 | . . 3 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β (βπ β π΄ (π β€ π β π β€ π) β§ βπ β π΄ (π β€ π β π β€ π)))) |
9 | 1, 8 | bitr4id 290 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (βπ β π΄ (π β€ π β π β€ π) β (π β€ π β§ π β€ π))) |
10 | hllat 38744 | . . 3 β’ (πΎ β HL β πΎ β Lat) | |
11 | 2, 3 | latasymb 18405 | . . 3 β’ ((πΎ β Lat β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
12 | 10, 11 | syl3an1 1160 | . 2 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β ((π β€ π β§ π β€ π) β π = π)) |
13 | 9, 12 | bitrd 279 | 1 β’ ((πΎ β HL β§ π β π΅ β§ π β π΅) β (βπ β π΄ (π β€ π β π β€ π) β π = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 class class class wbr 5141 βcfv 6536 Basecbs 17151 lecple 17211 Latclat 18394 Atomscatm 38644 HLchlt 38731 |
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 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-proset 18258 df-poset 18276 df-plt 18293 df-lub 18309 df-glb 18310 df-join 18311 df-meet 18312 df-p0 18388 df-lat 18395 df-clat 18462 df-oposet 38557 df-ol 38559 df-oml 38560 df-covers 38647 df-ats 38648 df-atl 38679 df-cvlat 38703 df-hlat 38732 |
This theorem is referenced by: lauteq 39477 ltrneq2 39530 |
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