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Mirrors > Home > MPE Home > Th. List > abelthlem7a | Structured version Visualization version GIF version |
Description: Lemma for abelth 25706. (Contributed by Mario Carneiro, 8-May-2015.) |
Ref | Expression |
---|---|
abelth.1 | β’ (π β π΄:β0βΆβ) |
abelth.2 | β’ (π β seq0( + , π΄) β dom β ) |
abelth.3 | β’ (π β π β β) |
abelth.4 | β’ (π β 0 β€ π) |
abelth.5 | β’ π = {π§ β β β£ (absβ(1 β π§)) β€ (π Β· (1 β (absβπ§)))} |
abelth.6 | β’ πΉ = (π₯ β π β¦ Ξ£π β β0 ((π΄βπ) Β· (π₯βπ))) |
abelth.7 | β’ (π β seq0( + , π΄) β 0) |
abelthlem6.1 | β’ (π β π β (π β {1})) |
Ref | Expression |
---|---|
abelthlem7a | β’ (π β (π β β β§ (absβ(1 β π)) β€ (π Β· (1 β (absβπ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abelthlem6.1 | . . 3 β’ (π β π β (π β {1})) | |
2 | 1 | eldifad 3910 | . 2 β’ (π β π β π) |
3 | oveq2 7345 | . . . . 5 β’ (π§ = π β (1 β π§) = (1 β π)) | |
4 | 3 | fveq2d 6829 | . . . 4 β’ (π§ = π β (absβ(1 β π§)) = (absβ(1 β π))) |
5 | fveq2 6825 | . . . . . 6 β’ (π§ = π β (absβπ§) = (absβπ)) | |
6 | 5 | oveq2d 7353 | . . . . 5 β’ (π§ = π β (1 β (absβπ§)) = (1 β (absβπ))) |
7 | 6 | oveq2d 7353 | . . . 4 β’ (π§ = π β (π Β· (1 β (absβπ§))) = (π Β· (1 β (absβπ)))) |
8 | 4, 7 | breq12d 5105 | . . 3 β’ (π§ = π β ((absβ(1 β π§)) β€ (π Β· (1 β (absβπ§))) β (absβ(1 β π)) β€ (π Β· (1 β (absβπ))))) |
9 | abelth.5 | . . 3 β’ π = {π§ β β β£ (absβ(1 β π§)) β€ (π Β· (1 β (absβπ§)))} | |
10 | 8, 9 | elrab2 3637 | . 2 β’ (π β π β (π β β β§ (absβ(1 β π)) β€ (π Β· (1 β (absβπ))))) |
11 | 2, 10 | sylib 217 | 1 β’ (π β (π β β β§ (absβ(1 β π)) β€ (π Β· (1 β (absβπ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 {crab 3403 β cdif 3895 {csn 4573 class class class wbr 5092 β¦ cmpt 5175 dom cdm 5620 βΆwf 6475 βcfv 6479 (class class class)co 7337 βcc 10970 βcr 10971 0cc0 10972 1c1 10973 + caddc 10975 Β· cmul 10977 β€ cle 11111 β cmin 11306 β0cn0 12334 seqcseq 13822 βcexp 13883 abscabs 15044 β cli 15292 Ξ£csu 15496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3404 df-v 3443 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-iota 6431 df-fv 6487 df-ov 7340 |
This theorem is referenced by: abelthlem7 25703 |
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