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Theorem cnrefiisplem 42108
Description: Lemma for cnrefiisp 42109 (some local definitions are used). (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypotheses
Ref Expression
cnrefiisplem.a (𝜑𝐴 ∈ ℂ)
cnrefiisplem.n (𝜑 → ¬ 𝐴 ∈ ℝ)
cnrefiisplem.b (𝜑𝐵 ∈ Fin)
cnrefiisplem.c 𝐶 = (ℝ ∪ 𝐵)
cnrefiisplem.d 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
cnrefiisplem.x 𝑋 = inf(𝐷, ℝ*, < )
Assertion
Ref Expression
cnrefiisplem (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
Distinct variable groups:   𝑦,𝐴,𝑥   𝑦,𝐵   𝑥,𝐶   𝑥,𝑋,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem cnrefiisplem
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simpr 487 . . . . . . 7 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 = (abs‘(ℑ‘𝐴)))
2 cnrefiisplem.a . . . . . . . . 9 (𝜑𝐴 ∈ ℂ)
3 cnrefiisplem.n . . . . . . . . 9 (𝜑 → ¬ 𝐴 ∈ ℝ)
42, 3absimnre 41751 . . . . . . . 8 (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
54adantr 483 . . . . . . 7 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
61, 5eqeltrd 2913 . . . . . 6 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
76adantlr 713 . . . . 5 (((𝜑𝑤𝐷) ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
8 simpll 765 . . . . . 6 (((𝜑𝑤𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝜑)
9 cnrefiisplem.d . . . . . . . . . . . 12 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
109eleq2i 2904 . . . . . . . . . . 11 (𝑤𝐷𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
1110biimpi 218 . . . . . . . . . 10 (𝑤𝐷𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
12 nelsn 4604 . . . . . . . . . 10 (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13 elunnel1 4125 . . . . . . . . . 10 ((𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}) ∧ ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))}) → 𝑤 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
1411, 12, 13syl2an 597 . . . . . . . . 9 ((𝑤𝐷𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
15 eliun 4922 . . . . . . . . 9 (𝑤 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦𝐴))})
1614, 15sylib 220 . . . . . . . 8 ((𝑤𝐷𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦𝐴))})
17 velsn 4582 . . . . . . . . 9 (𝑤 ∈ {(abs‘(𝑦𝐴))} ↔ 𝑤 = (abs‘(𝑦𝐴)))
1817rexbii 3247 . . . . . . . 8 (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)))
1916, 18sylib 220 . . . . . . 7 ((𝑤𝐷𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)))
2019adantll 712 . . . . . 6 (((𝜑𝑤𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)))
21 simpr 487 . . . . . . . 8 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑤 = (abs‘(𝑦𝐴)))
22 eldifi 4102 . . . . . . . . . . . 12 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
2322elin2d 4175 . . . . . . . . . . 11 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ ℂ)
2423ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑦 ∈ ℂ)
252ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝐴 ∈ ℂ)
2624, 25subcld 10996 . . . . . . . . 9 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (𝑦𝐴) ∈ ℂ)
27 eldifsni 4721 . . . . . . . . . . 11 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦𝐴)
2827ad2antlr 725 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑦𝐴)
2924, 25, 28subne0d 11005 . . . . . . . . 9 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (𝑦𝐴) ≠ 0)
3026, 29absrpcld 14807 . . . . . . . 8 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (abs‘(𝑦𝐴)) ∈ ℝ+)
3121, 30eqeltrd 2913 . . . . . . 7 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑤 ∈ ℝ+)
3231rexlimdva2 3287 . . . . . 6 (𝜑 → (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)) → 𝑤 ∈ ℝ+))
338, 20, 32sylc 65 . . . . 5 (((𝜑𝑤𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
347, 33pm2.61dane 3104 . . . 4 ((𝜑𝑤𝐷) → 𝑤 ∈ ℝ+)
3534ssd 41342 . . 3 (𝜑𝐷 ⊆ ℝ+)
36 cnrefiisplem.x . . . 4 𝑋 = inf(𝐷, ℝ*, < )
37 xrltso 12533 . . . . . 6 < Or ℝ*
3837a1i 11 . . . . 5 (𝜑 → < Or ℝ*)
39 snfi 8593 . . . . . . . 8 {(abs‘(ℑ‘𝐴))} ∈ Fin
4039a1i 11 . . . . . . 7 (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41 cnrefiisplem.b . . . . . . . . 9 (𝜑𝐵 ∈ Fin)
42 inss1 4204 . . . . . . . . . . 11 (𝐵 ∩ ℂ) ⊆ 𝐵
4342a1i 11 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
4443ssdifssd 4118 . . . . . . . . 9 (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
4541, 44ssfid 8740 . . . . . . . 8 (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46 snfi 8593 . . . . . . . . 9 {(abs‘(𝑦𝐴))} ∈ Fin
4746rgenw 3150 . . . . . . . 8 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin
48 iunfi 8811 . . . . . . . 8 ((((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin ∧ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin)
4945, 47, 48sylancl 588 . . . . . . 7 (𝜑 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin)
5040, 49unfid 41420 . . . . . 6 (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}) ∈ Fin)
519, 50eqeltrid 2917 . . . . 5 (𝜑𝐷 ∈ Fin)
52 fvex 6682 . . . . . . . . . 10 (abs‘(ℑ‘𝐴)) ∈ V
5352snid 4600 . . . . . . . . 9 (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54 elun1 4151 . . . . . . . . 9 ((abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))} → (abs‘(ℑ‘𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
5553, 54ax-mp 5 . . . . . . . 8 (abs‘(ℑ‘𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
5655, 9eleqtrri 2912 . . . . . . 7 (abs‘(ℑ‘𝐴)) ∈ 𝐷
5756a1i 11 . . . . . 6 (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
5857ne0d 4300 . . . . 5 (𝜑𝐷 ≠ ∅)
59 rpssxr 41755 . . . . . 6 + ⊆ ℝ*
6035, 59sstrdi 3978 . . . . 5 (𝜑𝐷 ⊆ ℝ*)
61 fiinfcl 8964 . . . . 5 (( < Or ℝ* ∧ (𝐷 ∈ Fin ∧ 𝐷 ≠ ∅ ∧ 𝐷 ⊆ ℝ*)) → inf(𝐷, ℝ*, < ) ∈ 𝐷)
6238, 51, 58, 60, 61syl13anc 1368 . . . 4 (𝜑 → inf(𝐷, ℝ*, < ) ∈ 𝐷)
6336, 62eqeltrid 2917 . . 3 (𝜑𝑋𝐷)
6435, 63sseldd 3967 . 2 (𝜑𝑋 ∈ ℝ+)
6535, 62sseldd 3967 . . . . . . . . . 10 (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
6665rpred 12430 . . . . . . . . 9 (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
6766adantr 483 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
682imcld 14553 . . . . . . . . . . 11 (𝜑 → (ℑ‘𝐴) ∈ ℝ)
6968recnd 10668 . . . . . . . . . 10 (𝜑 → (ℑ‘𝐴) ∈ ℂ)
7069adantr 483 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
7170abscld 14795 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72 recn 10626 . . . . . . . . . . 11 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
7372adantl 484 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
742adantr 483 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
7573, 74subcld 10996 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑦𝐴) ∈ ℂ)
7675abscld 14795 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(𝑦𝐴)) ∈ ℝ)
7760adantr 483 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78 infxrlb 12726 . . . . . . . . 9 ((𝐷 ⊆ ℝ* ∧ (abs‘(ℑ‘𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
7977, 56, 78sylancl 588 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
80 simpr 487 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
8174, 80absimlere 41754 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦𝐴)))
8267, 71, 76, 79, 81letrd 10796 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
8336, 82eqbrtrid 5100 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
8483ad4ant14 750 . . . . 5 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
85 cnrefiisplem.c . . . . . . . . 9 𝐶 = (ℝ ∪ 𝐵)
8685eleq2i 2904 . . . . . . . 8 (𝑦𝐶𝑦 ∈ (ℝ ∪ 𝐵))
87 elunnel1 4125 . . . . . . . 8 ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
8886, 87sylanb 583 . . . . . . 7 ((𝑦𝐶 ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
8988ad4ant24 752 . . . . . 6 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
9060ad2antrr 724 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝐷 ⊆ ℝ*)
91 simpr 487 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
92 simpll 765 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ ℂ)
9391, 92elind 4170 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94 nelsn 4604 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → ¬ 𝑦 ∈ {𝐴})
9594ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → ¬ 𝑦 ∈ {𝐴})
9693, 95eldifd 3946 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97 fvex 6682 . . . . . . . . . . . . . . 15 (abs‘(𝑦𝐴)) ∈ V
9897snid 4600 . . . . . . . . . . . . . 14 (abs‘(𝑦𝐴)) ∈ {(abs‘(𝑦𝐴))}
99 fvoveq1 7178 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (abs‘(𝑤𝐴)) = (abs‘(𝑦𝐴)))
10099sneqd 4578 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → {(abs‘(𝑤𝐴))} = {(abs‘(𝑦𝐴))})
101100eliuni 4924 . . . . . . . . . . . . . 14 ((𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) ∧ (abs‘(𝑦𝐴)) ∈ {(abs‘(𝑦𝐴))}) → (abs‘(𝑦𝐴)) ∈ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))})
10296, 98, 101sylancl 588 . . . . . . . . . . . . 13 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))})
103100cbviunv 4964 . . . . . . . . . . . . 13 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))} = 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}
104102, 103eleqtrdi 2923 . . . . . . . . . . . 12 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
105 elun2 4152 . . . . . . . . . . . 12 ((abs‘(𝑦𝐴)) ∈ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} → (abs‘(𝑦𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
106104, 105syl 17 . . . . . . . . . . 11 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
107106, 9eleqtrrdi 2924 . . . . . . . . . 10 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝐷)
108107adantll 712 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝐷)
109 infxrlb 12726 . . . . . . . . 9 ((𝐷 ⊆ ℝ* ∧ (abs‘(𝑦𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
11090, 108, 109syl2anc 586 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
11136, 110eqbrtrid 5100 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝑋 ≤ (abs‘(𝑦𝐴)))
112111adantllr 717 . . . . . 6 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝑋 ≤ (abs‘(𝑦𝐴)))
11389, 112syldan 593 . . . . 5 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
11484, 113pm2.61dan 811 . . . 4 (((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) → 𝑋 ≤ (abs‘(𝑦𝐴)))
115114ex 415 . . 3 ((𝜑𝑦𝐶) → ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴))))
116115ralrimiva 3182 . 2 (𝜑 → ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴))))
117 breq1 5068 . . . . 5 (𝑥 = 𝑋 → (𝑥 ≤ (abs‘(𝑦𝐴)) ↔ 𝑋 ≤ (abs‘(𝑦𝐴))))
118117imbi2d 343 . . . 4 (𝑥 = 𝑋 → (((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))) ↔ ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))))
119118ralbidv 3197 . . 3 (𝑥 = 𝑋 → (∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))) ↔ ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))))
120119rspcev 3622 . 2 ((𝑋 ∈ ℝ+ ∧ ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))) → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
12164, 116, 120syl2anc 586 1 (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1533  wcel 2110  wne 3016  wral 3138  wrex 3139  cdif 3932  cun 3933  cin 3934  wss 3935  c0 4290  {csn 4566   ciun 4918   class class class wbr 5065   Or wor 5472  cfv 6354  (class class class)co 7155  Fincfn 8508  infcinf 8904  cc 10534  cr 10535  *cxr 10673   < clt 10674  cle 10675  cmin 10869  +crp 12388  cim 14456  abscabs 14592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-sup 8905  df-inf 8906  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11699  df-3 11700  df-n0 11897  df-z 11981  df-uz 12243  df-rp 12389  df-seq 13369  df-exp 13429  df-cj 14457  df-re 14458  df-im 14459  df-sqrt 14593  df-abs 14594
This theorem is referenced by:  cnrefiisp  42109
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