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Theorem cnrefiisplem 45130
Description: Lemma for cnrefiisp 45131 (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 484 . . . . . . 7 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 = (abs‘(ℑ‘𝐴)))
2 cnrefiisplem.a . . . . . . . . 9 (𝜑𝐴 ∈ ℂ)
3 cnrefiisplem.n . . . . . . . . 9 (𝜑 → ¬ 𝐴 ∈ ℝ)
42, 3absimnre 44772 . . . . . . . 8 (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
54adantr 480 . . . . . . 7 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
61, 5eqeltrd 2828 . . . . . 6 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
76adantlr 714 . . . . 5 (((𝜑𝑤𝐷) ∧ 𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 ∈ ℝ+)
8 simpll 766 . . . . . 6 (((𝜑𝑤𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝜑)
9 cnrefiisplem.d . . . . . . . . . . . 12 𝐷 = ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
109eleq2i 2820 . . . . . . . . . . 11 (𝑤𝐷𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
1110biimpi 215 . . . . . . . . . 10 (𝑤𝐷𝑤 ∈ ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}))
12 nelsn 4664 . . . . . . . . . 10 (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13 elunnel1 4145 . . . . . . . . . 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 595 . . . . . . . . 9 ((𝑤𝐷𝑤 ≠ (abs‘(ℑ‘𝐴))) → 𝑤 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
15 eliun 4995 . . . . . . . . 9 (𝑤 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦𝐴))})
1614, 15sylib 217 . . . . . . . 8 ((𝑤𝐷𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦𝐴))})
17 velsn 4640 . . . . . . . . 9 (𝑤 ∈ {(abs‘(𝑦𝐴))} ↔ 𝑤 = (abs‘(𝑦𝐴)))
1817rexbii 3089 . . . . . . . 8 (∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 ∈ {(abs‘(𝑦𝐴))} ↔ ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)))
1916, 18sylib 217 . . . . . . 7 ((𝑤𝐷𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)))
2019adantll 713 . . . . . 6 (((𝜑𝑤𝐷) ∧ 𝑤 ≠ (abs‘(ℑ‘𝐴))) → ∃𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})𝑤 = (abs‘(𝑦𝐴)))
21 simpr 484 . . . . . . . 8 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑤 = (abs‘(𝑦𝐴)))
22 eldifi 4122 . . . . . . . . . . . 12 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
2322elin2d 4195 . . . . . . . . . . 11 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ ℂ)
2423ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑦 ∈ ℂ)
252ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝐴 ∈ ℂ)
2624, 25subcld 11587 . . . . . . . . 9 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (𝑦𝐴) ∈ ℂ)
27 eldifsni 4789 . . . . . . . . . . 11 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦𝐴)
2827ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑦𝐴)
2924, 25, 28subne0d 11596 . . . . . . . . 9 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (𝑦𝐴) ≠ 0)
3026, 29absrpcld 15413 . . . . . . . 8 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (abs‘(𝑦𝐴)) ∈ ℝ+)
3121, 30eqeltrd 2828 . . . . . . 7 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑤 ∈ ℝ+)
3231rexlimdva2 3152 . . . . . 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 3024 . . . 4 ((𝜑𝑤𝐷) → 𝑤 ∈ ℝ+)
3534ssd 44359 . . 3 (𝜑𝐷 ⊆ ℝ+)
36 cnrefiisplem.x . . . 4 𝑋 = inf(𝐷, ℝ*, < )
37 xrltso 13138 . . . . . 6 < Or ℝ*
3837a1i 11 . . . . 5 (𝜑 → < Or ℝ*)
39 snfi 9058 . . . . . . . 8 {(abs‘(ℑ‘𝐴))} ∈ Fin
4039a1i 11 . . . . . . 7 (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41 cnrefiisplem.b . . . . . . . . 9 (𝜑𝐵 ∈ Fin)
42 inss1 4224 . . . . . . . . . . 11 (𝐵 ∩ ℂ) ⊆ 𝐵
4342a1i 11 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
4443ssdifssd 4138 . . . . . . . . 9 (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
4541, 44ssfid 9281 . . . . . . . 8 (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46 snfi 9058 . . . . . . . . 9 {(abs‘(𝑦𝐴))} ∈ Fin
4746rgenw 3060 . . . . . . . 8 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin
48 iunfi 9354 . . . . . . . 8 ((((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin ∧ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin)
4945, 47, 48sylancl 585 . . . . . . 7 (𝜑 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin)
5040, 49unfid 44432 . . . . . 6 (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}) ∈ Fin)
519, 50eqeltrid 2832 . . . . 5 (𝜑𝐷 ∈ Fin)
52 fvex 6904 . . . . . . . . . 10 (abs‘(ℑ‘𝐴)) ∈ V
5352snid 4660 . . . . . . . . 9 (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54 elun1 4172 . . . . . . . . 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 2827 . . . . . . 7 (abs‘(ℑ‘𝐴)) ∈ 𝐷
5756a1i 11 . . . . . 6 (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
5857ne0d 4331 . . . . 5 (𝜑𝐷 ≠ ∅)
59 rpssxr 44776 . . . . . 6 + ⊆ ℝ*
6035, 59sstrdi 3990 . . . . 5 (𝜑𝐷 ⊆ ℝ*)
61 fiinfcl 9510 . . . . 5 (( < Or ℝ* ∧ (𝐷 ∈ Fin ∧ 𝐷 ≠ ∅ ∧ 𝐷 ⊆ ℝ*)) → inf(𝐷, ℝ*, < ) ∈ 𝐷)
6238, 51, 58, 60, 61syl13anc 1370 . . . 4 (𝜑 → inf(𝐷, ℝ*, < ) ∈ 𝐷)
6336, 62eqeltrid 2832 . . 3 (𝜑𝑋𝐷)
6435, 63sseldd 3979 . 2 (𝜑𝑋 ∈ ℝ+)
6535, 62sseldd 3979 . . . . . . . . . 10 (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
6665rpred 13034 . . . . . . . . 9 (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
6766adantr 480 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
682imcld 15160 . . . . . . . . . . 11 (𝜑 → (ℑ‘𝐴) ∈ ℝ)
6968recnd 11258 . . . . . . . . . 10 (𝜑 → (ℑ‘𝐴) ∈ ℂ)
7069adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
7170abscld 15401 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72 recn 11214 . . . . . . . . . . 11 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
7372adantl 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
742adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
7573, 74subcld 11587 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑦𝐴) ∈ ℂ)
7675abscld 15401 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(𝑦𝐴)) ∈ ℝ)
7760adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78 infxrlb 13331 . . . . . . . . 9 ((𝐷 ⊆ ℝ* ∧ (abs‘(ℑ‘𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
7977, 56, 78sylancl 585 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
80 simpr 484 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
8174, 80absimlere 44775 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦𝐴)))
8267, 71, 76, 79, 81letrd 11387 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
8336, 82eqbrtrid 5177 . . . . . 6 ((𝜑𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
8483ad4ant14 751 . . . . 5 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
85 cnrefiisplem.c . . . . . . . . 9 𝐶 = (ℝ ∪ 𝐵)
8685eleq2i 2820 . . . . . . . 8 (𝑦𝐶𝑦 ∈ (ℝ ∪ 𝐵))
87 elunnel1 4145 . . . . . . . 8 ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
8886, 87sylanb 580 . . . . . . 7 ((𝑦𝐶 ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
8988ad4ant24 753 . . . . . 6 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
9060ad2antrr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝐷 ⊆ ℝ*)
91 simpr 484 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
92 simpll 766 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ ℂ)
9391, 92elind 4190 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94 nelsn 4664 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → ¬ 𝑦 ∈ {𝐴})
9594ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → ¬ 𝑦 ∈ {𝐴})
9693, 95eldifd 3955 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97 fvex 6904 . . . . . . . . . . . . . . 15 (abs‘(𝑦𝐴)) ∈ V
9897snid 4660 . . . . . . . . . . . . . 14 (abs‘(𝑦𝐴)) ∈ {(abs‘(𝑦𝐴))}
99 fvoveq1 7437 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (abs‘(𝑤𝐴)) = (abs‘(𝑦𝐴)))
10099sneqd 4636 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → {(abs‘(𝑤𝐴))} = {(abs‘(𝑦𝐴))})
101100eliuni 4997 . . . . . . . . . . . . . 14 ((𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) ∧ (abs‘(𝑦𝐴)) ∈ {(abs‘(𝑦𝐴))}) → (abs‘(𝑦𝐴)) ∈ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))})
10296, 98, 101sylancl 585 . . . . . . . . . . . . 13 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))})
103100cbviunv 5037 . . . . . . . . . . . . 13 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))} = 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}
104102, 103eleqtrdi 2838 . . . . . . . . . . . 12 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
105 elun2 4173 . . . . . . . . . . . 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 2839 . . . . . . . . . 10 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝐷)
108107adantll 713 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝐷)
109 infxrlb 13331 . . . . . . . . 9 ((𝐷 ⊆ ℝ* ∧ (abs‘(𝑦𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
11090, 108, 109syl2anc 583 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
11136, 110eqbrtrid 5177 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝑋 ≤ (abs‘(𝑦𝐴)))
112111adantllr 718 . . . . . 6 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝑋 ≤ (abs‘(𝑦𝐴)))
11389, 112syldan 590 . . . . 5 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
11484, 113pm2.61dan 812 . . . 4 (((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) → 𝑋 ≤ (abs‘(𝑦𝐴)))
115114ex 412 . . 3 ((𝜑𝑦𝐶) → ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴))))
116115ralrimiva 3141 . 2 (𝜑 → ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴))))
117 breq1 5145 . . . . 5 (𝑥 = 𝑋 → (𝑥 ≤ (abs‘(𝑦𝐴)) ↔ 𝑋 ≤ (abs‘(𝑦𝐴))))
118117imbi2d 340 . . . 4 (𝑥 = 𝑋 → (((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))) ↔ ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))))
119118ralbidv 3172 . . 3 (𝑥 = 𝑋 → (∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))) ↔ ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))))
120119rspcev 3607 . 2 ((𝑋 ∈ ℝ+ ∧ ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))) → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
12164, 116, 120syl2anc 583 1 (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  cdif 3941  cun 3942  cin 3943  wss 3944  c0 4318  {csn 4624   ciun 4991   class class class wbr 5142   Or wor 5583  cfv 6542  (class class class)co 7414  Fincfn 8953  infcinf 9450  cc 11122  cr 11123  *cxr 11263   < clt 11264  cle 11265  cmin 11460  +crp 12992  cim 15063  abscabs 15199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-inf 9452  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-seq 13985  df-exp 14045  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201
This theorem is referenced by:  cnrefiisp  45131
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