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Theorem abelthlem2 25807
Description: Lemma for abelth 25816. The peculiar region 𝑆, known as a Stolz angle , is a teardrop-shaped subset of the closed unit ball containing 1. Indeed, except for 1 itself, the rest of the Stolz angle is enclosed in the open unit ball. (Contributed by Mario Carneiro, 31-Mar-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‘§)))}
Assertion
Ref Expression
abelthlem2 (πœ‘ β†’ (1 ∈ 𝑆 ∧ (𝑆 βˆ– {1}) βŠ† (0(ballβ€˜(abs ∘ βˆ’ ))1)))
Distinct variable groups:   𝑧,𝑀   𝑧,𝐴
Allowed substitution hints:   πœ‘(𝑧)   𝑆(𝑧)

Proof of Theorem abelthlem2
StepHypRef Expression
1 abelth.3 . 2 (πœ‘ β†’ 𝑀 ∈ ℝ)
2 abelth.4 . 2 (πœ‘ β†’ 0 ≀ 𝑀)
3 1cnd 11155 . . . 4 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ 1 ∈ β„‚)
4 0le0 12259 . . . . 5 0 ≀ 0
5 simpl 484 . . . . . . 7 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ 𝑀 ∈ ℝ)
65recnd 11188 . . . . . 6 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ 𝑀 ∈ β„‚)
76mul01d 11359 . . . . 5 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ (𝑀 Β· 0) = 0)
84, 7breqtrrid 5144 . . . 4 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ 0 ≀ (𝑀 Β· 0))
9 oveq2 7366 . . . . . . . 8 (𝑧 = 1 β†’ (1 βˆ’ 𝑧) = (1 βˆ’ 1))
10 1m1e0 12230 . . . . . . . 8 (1 βˆ’ 1) = 0
119, 10eqtrdi 2789 . . . . . . 7 (𝑧 = 1 β†’ (1 βˆ’ 𝑧) = 0)
1211abs00bd 15182 . . . . . 6 (𝑧 = 1 β†’ (absβ€˜(1 βˆ’ 𝑧)) = 0)
13 fveq2 6843 . . . . . . . . . 10 (𝑧 = 1 β†’ (absβ€˜π‘§) = (absβ€˜1))
14 abs1 15188 . . . . . . . . . 10 (absβ€˜1) = 1
1513, 14eqtrdi 2789 . . . . . . . . 9 (𝑧 = 1 β†’ (absβ€˜π‘§) = 1)
1615oveq2d 7374 . . . . . . . 8 (𝑧 = 1 β†’ (1 βˆ’ (absβ€˜π‘§)) = (1 βˆ’ 1))
1716, 10eqtrdi 2789 . . . . . . 7 (𝑧 = 1 β†’ (1 βˆ’ (absβ€˜π‘§)) = 0)
1817oveq2d 7374 . . . . . 6 (𝑧 = 1 β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) = (𝑀 Β· 0))
1912, 18breq12d 5119 . . . . 5 (𝑧 = 1 β†’ ((absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) ↔ 0 ≀ (𝑀 Β· 0)))
20 abelth.5 . . . . 5 𝑆 = {𝑧 ∈ β„‚ ∣ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))}
2119, 20elrab2 3649 . . . 4 (1 ∈ 𝑆 ↔ (1 ∈ β„‚ ∧ 0 ≀ (𝑀 Β· 0)))
223, 8, 21sylanbrc 584 . . 3 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ 1 ∈ 𝑆)
23 velsn 4603 . . . . . . . . . 10 (𝑧 ∈ {1} ↔ 𝑧 = 1)
2423necon3bbii 2988 . . . . . . . . 9 (Β¬ 𝑧 ∈ {1} ↔ 𝑧 β‰  1)
25 simprll 778 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 𝑧 ∈ β„‚)
26 0cn 11152 . . . . . . . . . . . . . . 15 0 ∈ β„‚
27 eqid 2733 . . . . . . . . . . . . . . . 16 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
2827cnmetdval 24150 . . . . . . . . . . . . . . 15 ((𝑧 ∈ β„‚ ∧ 0 ∈ β„‚) β†’ (𝑧(abs ∘ βˆ’ )0) = (absβ€˜(𝑧 βˆ’ 0)))
2925, 26, 28sylancl 587 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑧(abs ∘ βˆ’ )0) = (absβ€˜(𝑧 βˆ’ 0)))
3025subid1d 11506 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑧 βˆ’ 0) = 𝑧)
3130fveq2d 6847 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜(𝑧 βˆ’ 0)) = (absβ€˜π‘§))
3229, 31eqtrd 2773 . . . . . . . . . . . . 13 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑧(abs ∘ βˆ’ )0) = (absβ€˜π‘§))
3325abscld 15327 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜π‘§) ∈ ℝ)
34 1red 11161 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 1 ∈ ℝ)
35 1re 11160 . . . . . . . . . . . . . . . . . . . . 21 1 ∈ ℝ
36 resubcl 11470 . . . . . . . . . . . . . . . . . . . . 21 (((absβ€˜π‘§) ∈ ℝ ∧ 1 ∈ ℝ) β†’ ((absβ€˜π‘§) βˆ’ 1) ∈ ℝ)
3733, 35, 36sylancl 587 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((absβ€˜π‘§) βˆ’ 1) ∈ ℝ)
38 ax-1cn 11114 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ β„‚
39 subcl 11405 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (1 βˆ’ 𝑧) ∈ β„‚)
4038, 25, 39sylancr 588 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (1 βˆ’ 𝑧) ∈ β„‚)
4140abscld 15327 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜(1 βˆ’ 𝑧)) ∈ ℝ)
42 simpll 766 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 𝑀 ∈ ℝ)
43 resubcl 11470 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ ℝ ∧ (absβ€˜π‘§) ∈ ℝ) β†’ (1 βˆ’ (absβ€˜π‘§)) ∈ ℝ)
4435, 33, 43sylancr 588 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (1 βˆ’ (absβ€˜π‘§)) ∈ ℝ)
4542, 44remulcld 11190 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) ∈ ℝ)
4614oveq2i 7369 . . . . . . . . . . . . . . . . . . . . . 22 ((absβ€˜π‘§) βˆ’ (absβ€˜1)) = ((absβ€˜π‘§) βˆ’ 1)
47 abs2dif 15223 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜1)) ≀ (absβ€˜(𝑧 βˆ’ 1)))
4825, 38, 47sylancl 587 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜1)) ≀ (absβ€˜(𝑧 βˆ’ 1)))
4946, 48eqbrtrrid 5142 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((absβ€˜π‘§) βˆ’ 1) ≀ (absβ€˜(𝑧 βˆ’ 1)))
50 abssub 15217 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ (absβ€˜(𝑧 βˆ’ 1)) = (absβ€˜(1 βˆ’ 𝑧)))
5125, 38, 50sylancl 587 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜(𝑧 βˆ’ 1)) = (absβ€˜(1 βˆ’ 𝑧)))
5249, 51breqtrd 5132 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((absβ€˜π‘§) βˆ’ 1) ≀ (absβ€˜(1 βˆ’ 𝑧)))
53 simprlr 779 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))))
5437, 41, 45, 52, 53letrd 11317 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((absβ€˜π‘§) βˆ’ 1) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))))
5533, 34, 45lesubaddd 11757 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (((absβ€˜π‘§) βˆ’ 1) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) ↔ (absβ€˜π‘§) ≀ ((𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) + 1)))
5654, 55mpbid 231 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜π‘§) ≀ ((𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) + 1))
576adantr 482 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 𝑀 ∈ β„‚)
58 1cnd 11155 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 1 ∈ β„‚)
5942, 33remulcld 11190 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· (absβ€˜π‘§)) ∈ ℝ)
6059recnd 11188 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· (absβ€˜π‘§)) ∈ β„‚)
6157, 58, 60addsubd 11538 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 + 1) βˆ’ (𝑀 Β· (absβ€˜π‘§))) = ((𝑀 βˆ’ (𝑀 Β· (absβ€˜π‘§))) + 1))
6233recnd 11188 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜π‘§) ∈ β„‚)
6357, 58, 62subdid 11616 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) = ((𝑀 Β· 1) βˆ’ (𝑀 Β· (absβ€˜π‘§))))
6457mulid1d 11177 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· 1) = 𝑀)
6564oveq1d 7373 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 Β· 1) βˆ’ (𝑀 Β· (absβ€˜π‘§))) = (𝑀 βˆ’ (𝑀 Β· (absβ€˜π‘§))))
6663, 65eqtrd 2773 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) = (𝑀 βˆ’ (𝑀 Β· (absβ€˜π‘§))))
6766oveq1d 7373 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) + 1) = ((𝑀 βˆ’ (𝑀 Β· (absβ€˜π‘§))) + 1))
6861, 67eqtr4d 2776 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 + 1) βˆ’ (𝑀 Β· (absβ€˜π‘§))) = ((𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) + 1))
6956, 68breqtrrd 5134 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜π‘§) ≀ ((𝑀 + 1) βˆ’ (𝑀 Β· (absβ€˜π‘§))))
70 peano2re 11333 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℝ β†’ (𝑀 + 1) ∈ ℝ)
7142, 70syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 + 1) ∈ ℝ)
7259, 33, 71leaddsub2d 11762 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (((𝑀 Β· (absβ€˜π‘§)) + (absβ€˜π‘§)) ≀ (𝑀 + 1) ↔ (absβ€˜π‘§) ≀ ((𝑀 + 1) βˆ’ (𝑀 Β· (absβ€˜π‘§)))))
7369, 72mpbird 257 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 Β· (absβ€˜π‘§)) + (absβ€˜π‘§)) ≀ (𝑀 + 1))
7457, 62adddirp1d 11186 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 + 1) Β· (absβ€˜π‘§)) = ((𝑀 Β· (absβ€˜π‘§)) + (absβ€˜π‘§)))
7571recnd 11188 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 + 1) ∈ β„‚)
7675mulid1d 11177 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 + 1) Β· 1) = (𝑀 + 1))
7773, 74, 763brtr4d 5138 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((𝑀 + 1) Β· (absβ€˜π‘§)) ≀ ((𝑀 + 1) Β· 1))
78 0red 11163 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 0 ∈ ℝ)
79 simplr 768 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 0 ≀ 𝑀)
8042ltp1d 12090 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 𝑀 < (𝑀 + 1))
8178, 42, 71, 79, 80lelttrd 11318 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 0 < (𝑀 + 1))
82 lemul2 12013 . . . . . . . . . . . . . . . 16 (((absβ€˜π‘§) ∈ ℝ ∧ 1 ∈ ℝ ∧ ((𝑀 + 1) ∈ ℝ ∧ 0 < (𝑀 + 1))) β†’ ((absβ€˜π‘§) ≀ 1 ↔ ((𝑀 + 1) Β· (absβ€˜π‘§)) ≀ ((𝑀 + 1) Β· 1)))
8333, 34, 71, 81, 82syl112anc 1375 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((absβ€˜π‘§) ≀ 1 ↔ ((𝑀 + 1) Β· (absβ€˜π‘§)) ≀ ((𝑀 + 1) Β· 1)))
8477, 83mpbird 257 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜π‘§) ≀ 1)
8541, 45, 53lensymd 11311 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ Β¬ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) < (absβ€˜(1 βˆ’ 𝑧)))
867adantr 482 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· 0) = 0)
87 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 𝑧 β‰  1)
8887necomd 2996 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 1 β‰  𝑧)
89 subeq0 11432 . . . . . . . . . . . . . . . . . . . . . 22 ((1 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ ((1 βˆ’ 𝑧) = 0 ↔ 1 = 𝑧))
9089necon3bid 2985 . . . . . . . . . . . . . . . . . . . . 21 ((1 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ ((1 βˆ’ 𝑧) β‰  0 ↔ 1 β‰  𝑧))
9138, 25, 90sylancr 588 . . . . . . . . . . . . . . . . . . . 20 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((1 βˆ’ 𝑧) β‰  0 ↔ 1 β‰  𝑧))
9288, 91mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (1 βˆ’ 𝑧) β‰  0)
93 absgt0 15215 . . . . . . . . . . . . . . . . . . . 20 ((1 βˆ’ 𝑧) ∈ β„‚ β†’ ((1 βˆ’ 𝑧) β‰  0 ↔ 0 < (absβ€˜(1 βˆ’ 𝑧))))
9440, 93syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ ((1 βˆ’ 𝑧) β‰  0 ↔ 0 < (absβ€˜(1 βˆ’ 𝑧))))
9592, 94mpbid 231 . . . . . . . . . . . . . . . . . 18 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 0 < (absβ€˜(1 βˆ’ 𝑧)))
9686, 95eqbrtrd 5128 . . . . . . . . . . . . . . . . 17 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑀 Β· 0) < (absβ€˜(1 βˆ’ 𝑧)))
97 oveq2 7366 . . . . . . . . . . . . . . . . . . . 20 (1 = (absβ€˜π‘§) β†’ (1 βˆ’ 1) = (1 βˆ’ (absβ€˜π‘§)))
9810, 97eqtr3id 2787 . . . . . . . . . . . . . . . . . . 19 (1 = (absβ€˜π‘§) β†’ 0 = (1 βˆ’ (absβ€˜π‘§)))
9998oveq2d 7374 . . . . . . . . . . . . . . . . . 18 (1 = (absβ€˜π‘§) β†’ (𝑀 Β· 0) = (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))))
10099breq1d 5116 . . . . . . . . . . . . . . . . 17 (1 = (absβ€˜π‘§) β†’ ((𝑀 Β· 0) < (absβ€˜(1 βˆ’ 𝑧)) ↔ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) < (absβ€˜(1 βˆ’ 𝑧))))
10196, 100syl5ibcom 244 . . . . . . . . . . . . . . . 16 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (1 = (absβ€˜π‘§) β†’ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) < (absβ€˜(1 βˆ’ 𝑧))))
102101necon3bd 2954 . . . . . . . . . . . . . . 15 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (Β¬ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))) < (absβ€˜(1 βˆ’ 𝑧)) β†’ 1 β‰  (absβ€˜π‘§)))
10385, 102mpd 15 . . . . . . . . . . . . . 14 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 1 β‰  (absβ€˜π‘§))
10433, 34, 84, 103leneltd 11314 . . . . . . . . . . . . 13 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (absβ€˜π‘§) < 1)
10532, 104eqbrtrd 5128 . . . . . . . . . . . 12 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑧(abs ∘ βˆ’ )0) < 1)
106 cnxmet 24152 . . . . . . . . . . . . . 14 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
107 1xr 11219 . . . . . . . . . . . . . 14 1 ∈ ℝ*
108 elbl3 23761 . . . . . . . . . . . . . 14 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 1 ∈ ℝ*) ∧ (0 ∈ β„‚ ∧ 𝑧 ∈ β„‚)) β†’ (𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1) ↔ (𝑧(abs ∘ βˆ’ )0) < 1))
109106, 107, 108mpanl12 701 . . . . . . . . . . . . 13 ((0 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1) ↔ (𝑧(abs ∘ βˆ’ )0) < 1))
11026, 25, 109sylancr 588 . . . . . . . . . . . 12 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ (𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1) ↔ (𝑧(abs ∘ βˆ’ )0) < 1))
111105, 110mpbird 257 . . . . . . . . . . 11 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ ((𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) ∧ 𝑧 β‰  1)) β†’ 𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1))
112111expr 458 . . . . . . . . . 10 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ (𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§))))) β†’ (𝑧 β‰  1 β†’ 𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
1131123impb 1116 . . . . . . . . 9 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ 𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) β†’ (𝑧 β‰  1 β†’ 𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
11424, 113biimtrid 241 . . . . . . . 8 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ 𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) β†’ (Β¬ 𝑧 ∈ {1} β†’ 𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
115114orrd 862 . . . . . . 7 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ 𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) β†’ (𝑧 ∈ {1} ∨ 𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
116 elun 4109 . . . . . . 7 (𝑧 ∈ ({1} βˆͺ (0(ballβ€˜(abs ∘ βˆ’ ))1)) ↔ (𝑧 ∈ {1} ∨ 𝑧 ∈ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
117115, 116sylibr 233 . . . . . 6 (((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) ∧ 𝑧 ∈ β„‚ ∧ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))) β†’ 𝑧 ∈ ({1} βˆͺ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
118117rabssdv 4033 . . . . 5 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ {𝑧 ∈ β„‚ ∣ (absβ€˜(1 βˆ’ 𝑧)) ≀ (𝑀 Β· (1 βˆ’ (absβ€˜π‘§)))} βŠ† ({1} βˆͺ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
11920, 118eqsstrid 3993 . . . 4 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ 𝑆 βŠ† ({1} βˆͺ (0(ballβ€˜(abs ∘ βˆ’ ))1)))
120 ssundif 4446 . . . 4 (𝑆 βŠ† ({1} βˆͺ (0(ballβ€˜(abs ∘ βˆ’ ))1)) ↔ (𝑆 βˆ– {1}) βŠ† (0(ballβ€˜(abs ∘ βˆ’ ))1))
121119, 120sylib 217 . . 3 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ (𝑆 βˆ– {1}) βŠ† (0(ballβ€˜(abs ∘ βˆ’ ))1))
12222, 121jca 513 . 2 ((𝑀 ∈ ℝ ∧ 0 ≀ 𝑀) β†’ (1 ∈ 𝑆 ∧ (𝑆 βˆ– {1}) βŠ† (0(ballβ€˜(abs ∘ βˆ’ ))1)))
1231, 2, 122syl2anc 585 1 (πœ‘ β†’ (1 ∈ 𝑆 ∧ (𝑆 βˆ– {1}) βŠ† (0(ballβ€˜(abs ∘ βˆ’ ))1)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  {crab 3406   βˆ– cdif 3908   βˆͺ cun 3909   βŠ† wss 3911  {csn 4587   class class class wbr 5106  dom cdm 5634   ∘ ccom 5638  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  β„cr 11055  0cc0 11056  1c1 11057   + caddc 11059   Β· cmul 11061  β„*cxr 11193   < clt 11194   ≀ cle 11195   βˆ’ cmin 11390  β„•0cn0 12418  seqcseq 13912  abscabs 15125   ⇝ cli 15372  βˆžMetcxmet 20797  ballcbl 20799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-xadd 13039  df-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807
This theorem is referenced by:  abelthlem3  25808  abelthlem6  25811  abelthlem7  25813  abelthlem8  25814  abelthlem9  25815  abelth  25816
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