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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 32196 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 3110 | . . 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 38871 | . . . 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 38871 | . . . . 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 1124 | . . . 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 631 | . . 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 38835 | . . 3 β’ (πΎ β HL β πΎ β Lat) | |
11 | 2, 3 | latasymb 18434 | . . 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 1161 | . 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 1085 = wceq 1534 β wcel 2099 βwral 3058 class class class wbr 5148 βcfv 6548 Basecbs 17180 lecple 17240 Latclat 18423 Atomscatm 38735 HLchlt 38822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-lat 18424 df-clat 18491 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 |
This theorem is referenced by: lauteq 39568 ltrneq2 39621 |
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