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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartlemelr | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 32851. (Contributed by Thierry Arnoux, 8-Aug-2018.) |
Ref | Expression |
---|---|
eulerpartlems.r | β’ π = {π β£ (β‘π β β) β Fin} |
eulerpartlems.s | β’ π = (π β ((β0 βm β) β© π ) β¦ Ξ£π β β ((πβπ) Β· π)) |
Ref | Expression |
---|---|
eulerpartlemelr | β’ (π΄ β ((β0 βm β) β© π ) β (π΄:ββΆβ0 β§ (β‘π΄ β β) β Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 4186 | . . . 4 β’ ((β0 βm β) β© π ) β (β0 βm β) | |
2 | 1 | sseli 3938 | . . 3 β’ (π΄ β ((β0 βm β) β© π ) β π΄ β (β0 βm β)) |
3 | elmapi 8783 | . . 3 β’ (π΄ β (β0 βm β) β π΄:ββΆβ0) | |
4 | 2, 3 | syl 17 | . 2 β’ (π΄ β ((β0 βm β) β© π ) β π΄:ββΆβ0) |
5 | inss2 4187 | . . . 4 β’ ((β0 βm β) β© π ) β π | |
6 | 5 | sseli 3938 | . . 3 β’ (π΄ β ((β0 βm β) β© π ) β π΄ β π ) |
7 | cnveq 5827 | . . . . . 6 β’ (π = π΄ β β‘π = β‘π΄) | |
8 | 7 | imaeq1d 6010 | . . . . 5 β’ (π = π΄ β (β‘π β β) = (β‘π΄ β β)) |
9 | 8 | eleq1d 2822 | . . . 4 β’ (π = π΄ β ((β‘π β β) β Fin β (β‘π΄ β β) β Fin)) |
10 | eulerpartlems.r | . . . 4 β’ π = {π β£ (β‘π β β) β Fin} | |
11 | 9, 10 | elab2g 3630 | . . 3 β’ (π΄ β ((β0 βm β) β© π ) β (π΄ β π β (β‘π΄ β β) β Fin)) |
12 | 6, 11 | mpbid 231 | . 2 β’ (π΄ β ((β0 βm β) β© π ) β (β‘π΄ β β) β Fin) |
13 | 4, 12 | jca 512 | 1 β’ (π΄ β ((β0 βm β) β© π ) β (π΄:ββΆβ0 β§ (β‘π΄ β β) β Fin)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {cab 2713 β© cin 3907 β¦ cmpt 5186 β‘ccnv 5630 β cima 5634 βΆwf 6489 βcfv 6493 (class class class)co 7353 βm cmap 8761 Fincfn 8879 Β· cmul 11052 βcn 12149 β0cn0 12409 Ξ£csu 15562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-map 8763 |
This theorem is referenced by: eulerpartlemsv2 32827 eulerpartlemsf 32828 eulerpartlems 32829 eulerpartlemsv3 32830 eulerpartlemgc 32831 |
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