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Mirrors > Home > MPE Home > Th. List > Mathboxes > eulerpartleme | Structured version Visualization version GIF version |
Description: Lemma for eulerpart 34055. (Contributed by Mario Carneiro, 26-Jan-2015.) |
Ref | Expression |
---|---|
eulerpart.p | β’ π = {π β (β0 βm β) β£ ((β‘π β β) β Fin β§ Ξ£π β β ((πβπ) Β· π) = π)} |
Ref | Expression |
---|---|
eulerpartleme | β’ (π΄ β π β (π΄:ββΆβ0 β§ (β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0ex 12503 | . . . 4 β’ β0 β V | |
2 | nnex 12243 | . . . 4 β’ β β V | |
3 | 1, 2 | elmap 8883 | . . 3 β’ (π΄ β (β0 βm β) β π΄:ββΆβ0) |
4 | 3 | anbi1i 622 | . 2 β’ ((π΄ β (β0 βm β) β§ ((β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π)) β (π΄:ββΆβ0 β§ ((β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π))) |
5 | cnveq 5871 | . . . . . 6 β’ (π = π΄ β β‘π = β‘π΄) | |
6 | 5 | imaeq1d 6058 | . . . . 5 β’ (π = π΄ β (β‘π β β) = (β‘π΄ β β)) |
7 | 6 | eleq1d 2810 | . . . 4 β’ (π = π΄ β ((β‘π β β) β Fin β (β‘π΄ β β) β Fin)) |
8 | fveq1 6889 | . . . . . . 7 β’ (π = π΄ β (πβπ) = (π΄βπ)) | |
9 | 8 | oveq1d 7428 | . . . . . 6 β’ (π = π΄ β ((πβπ) Β· π) = ((π΄βπ) Β· π)) |
10 | 9 | sumeq2sdv 15677 | . . . . 5 β’ (π = π΄ β Ξ£π β β ((πβπ) Β· π) = Ξ£π β β ((π΄βπ) Β· π)) |
11 | 10 | eqeq1d 2727 | . . . 4 β’ (π = π΄ β (Ξ£π β β ((πβπ) Β· π) = π β Ξ£π β β ((π΄βπ) Β· π) = π)) |
12 | 7, 11 | anbi12d 630 | . . 3 β’ (π = π΄ β (((β‘π β β) β Fin β§ Ξ£π β β ((πβπ) Β· π) = π) β ((β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π))) |
13 | eulerpart.p | . . 3 β’ π = {π β (β0 βm β) β£ ((β‘π β β) β Fin β§ Ξ£π β β ((πβπ) Β· π) = π)} | |
14 | 12, 13 | elrab2 3679 | . 2 β’ (π΄ β π β (π΄ β (β0 βm β) β§ ((β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π))) |
15 | 3anass 1092 | . 2 β’ ((π΄:ββΆβ0 β§ (β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π) β (π΄:ββΆβ0 β§ ((β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π))) | |
16 | 4, 14, 15 | 3bitr4i 302 | 1 β’ (π΄ β π β (π΄:ββΆβ0 β§ (β‘π΄ β β) β Fin β§ Ξ£π β β ((π΄βπ) Β· π) = π)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 {crab 3419 β‘ccnv 5672 β cima 5676 βΆwf 6539 βcfv 6543 (class class class)co 7413 βm cmap 8838 Fincfn 8957 Β· cmul 11138 βcn 12237 β0cn0 12497 Ξ£csu 15659 |
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-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-nel 3037 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-pss 3961 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-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 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-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-seq 13994 df-sum 15660 |
This theorem is referenced by: eulerpartlemv 34037 eulerpartlemd 34039 eulerpartlemb 34041 eulerpartlemn 34054 |
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