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Theorem abelthlem7a 25702
Description: Lemma for abelth 25706. (Contributed by Mario Carneiro, 8-May-2015.)
Hypotheses
Ref Expression
abelth.1 (πœ‘ β†’ 𝐴:β„•0βŸΆβ„‚)
abelth.2 (πœ‘ β†’ seq0( + , 𝐴) ∈ dom ⇝ )
abelth.3 (πœ‘ β†’ 𝑀 ∈ ℝ)
abelth.4 (πœ‘ β†’ 0 ≀ 𝑀)
abelth.5 𝑆 = {𝑧 ∈ β„‚ ∣ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))}
abelth.6 𝐹 = (π‘₯ ∈ 𝑆 ↦ Σ𝑛 ∈ β„•0 ((π΄β€˜π‘›) Β· (π‘₯↑𝑛)))
abelth.7 (πœ‘ β†’ seq0( + , 𝐴) ⇝ 0)
abelthlem6.1 (πœ‘ β†’ 𝑋 ∈ (𝑆 βˆ– {1}))
Assertion
Ref Expression
abelthlem7a (πœ‘ β†’ (𝑋 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑋)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘‹)))))
Distinct variable groups:   π‘₯,𝑛,𝑧,𝑀   𝑛,𝑋,π‘₯,𝑧   𝐴,𝑛,π‘₯,𝑧   πœ‘,𝑛,π‘₯   𝑆,𝑛,π‘₯
Allowed substitution hints:   πœ‘(𝑧)   𝑆(𝑧)   𝐹(π‘₯,𝑧,𝑛)

Proof of Theorem abelthlem7a
StepHypRef Expression
1 abelthlem6.1 . . 3 (πœ‘ β†’ 𝑋 ∈ (𝑆 βˆ– {1}))
21eldifad 3910 . 2 (πœ‘ β†’ 𝑋 ∈ 𝑆)
3 oveq2 7345 . . . . 5 (𝑧 = 𝑋 β†’ (1 βˆ’ 𝑧) = (1 βˆ’ 𝑋))
43fveq2d 6829 . . . 4 (𝑧 = 𝑋 β†’ (absβ€˜(1 βˆ’ 𝑧)) = (absβ€˜(1 βˆ’ 𝑋)))
5 fveq2 6825 . . . . . 6 (𝑧 = 𝑋 β†’ (absβ€˜π‘§) = (absβ€˜π‘‹))
65oveq2d 7353 . . . . 5 (𝑧 = 𝑋 β†’ (1 βˆ’ (absβ€˜π‘§)) = (1 βˆ’ (absβ€˜π‘‹)))
76oveq2d 7353 . . . 4 (𝑧 = 𝑋 β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) = (𝑀 Β· (1 βˆ’ (absβ€˜π‘‹))))
84, 7breq12d 5105 . . 3 (𝑧 = 𝑋 β†’ ((absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) ↔ (absβ€˜(1 βˆ’ 𝑋)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘‹)))))
9 abelth.5 . . 3 𝑆 = {𝑧 ∈ β„‚ ∣ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))}
108, 9elrab2 3637 . 2 (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑋)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘‹)))))
112, 10sylib 217 1 (πœ‘ β†’ (𝑋 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑋)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘‹)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {crab 3403   βˆ– cdif 3895  {csn 4573   class class class wbr 5092   ↦ cmpt 5175  dom cdm 5620  βŸΆwf 6475  β€˜cfv 6479  (class class class)co 7337  β„‚cc 10970  β„cr 10971  0cc0 10972  1c1 10973   + caddc 10975   Β· cmul 10977   ≀ cle 11111   βˆ’ cmin 11306  β„•0cn0 12334  seqcseq 13822  β†‘cexp 13883  abscabs 15044   ⇝ 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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