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Mirrors > Home > MPE Home > Th. List > Mathboxes > paddasslem11 | Structured version Visualization version GIF version |
Description: Lemma for paddass 39363. The case when π = π§. (Contributed by NM, 11-Jan-2012.) |
Ref | Expression |
---|---|
paddasslem.l | β’ β€ = (leβπΎ) |
paddasslem.j | β’ β¨ = (joinβπΎ) |
paddasslem.a | β’ π΄ = (AtomsβπΎ) |
paddasslem.p | β’ + = (+πβπΎ) |
Ref | Expression |
---|---|
paddasslem11 | β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β ((π + π) + π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplll 773 | . . 3 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β πΎ β HL) | |
2 | simplr3 1214 | . . 3 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β π΄) | |
3 | simplr1 1212 | . . . 4 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β π΄) | |
4 | simplr2 1213 | . . . 4 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β π΄) | |
5 | paddasslem.a | . . . . 5 β’ π΄ = (AtomsβπΎ) | |
6 | paddasslem.p | . . . . 5 β’ + = (+πβπΎ) | |
7 | 5, 6 | paddssat 39339 | . . . 4 β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π + π) β π΄) |
8 | 1, 3, 4, 7 | syl3anc 1368 | . . 3 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β (π + π) β π΄) |
9 | 5, 6 | sspadd2 39341 | . . 3 β’ ((πΎ β HL β§ π β π΄ β§ (π + π) β π΄) β π β ((π + π) + π)) |
10 | 1, 2, 8, 9 | syl3anc 1368 | . 2 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β ((π + π) + π)) |
11 | simpllr 774 | . . 3 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π = π§) | |
12 | simpr 483 | . . 3 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π§ β π) | |
13 | 11, 12 | eqeltrd 2825 | . 2 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β π) |
14 | 10, 13 | sseldd 3974 | 1 β’ ((((πΎ β HL β§ π = π§) β§ (π β π΄ β§ π β π΄ β§ π β π΄)) β§ π§ β π) β π β ((π + π) + π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3941 βcfv 6543 (class class class)co 7413 lecple 17234 joincjn 18297 Atomscatm 38787 HLchlt 38874 +πcpadd 39320 |
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 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7987 df-2nd 7988 df-padd 39321 |
This theorem is referenced by: paddasslem14 39358 |
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