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Theorem abelthlem7a 25956
Description: Lemma for abelth 25960. (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 3960 . 2 (πœ‘ β†’ 𝑋 ∈ 𝑆)
3 oveq2 7419 . . . . 5 (𝑧 = 𝑋 β†’ (1 βˆ’ 𝑧) = (1 βˆ’ 𝑋))
43fveq2d 6895 . . . 4 (𝑧 = 𝑋 β†’ (absβ€˜(1 βˆ’ 𝑧)) = (absβ€˜(1 βˆ’ 𝑋)))
5 fveq2 6891 . . . . . 6 (𝑧 = 𝑋 β†’ (absβ€˜π‘§) = (absβ€˜π‘‹))
65oveq2d 7427 . . . . 5 (𝑧 = 𝑋 β†’ (1 βˆ’ (absβ€˜π‘§)) = (1 βˆ’ (absβ€˜π‘‹)))
76oveq2d 7427 . . . 4 (𝑧 = 𝑋 β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) = (𝑀 Β· (1 βˆ’ (absβ€˜π‘‹))))
84, 7breq12d 5161 . . 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 3686 . 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 1541   ∈ wcel 2106  {crab 3432   βˆ– cdif 3945  {csn 4628   class class class wbr 5148   ↦ cmpt 5231  dom cdm 5676  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   Β· cmul 11117   ≀ cle 11251   βˆ’ cmin 11446  β„•0cn0 12474  seqcseq 13968  β†‘cexp 14029  abscabs 15183   ⇝ cli 15430  Ξ£csu 15634
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-ext 2703
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-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414
This theorem is referenced by:  abelthlem7  25957
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