Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cnrefiisplem Structured version   Visualization version   GIF version

Theorem cnrefiisplem 44156
Description: Lemma for cnrefiisp 44157 (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 486 . . . . . . 7 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → 𝑤 = (abs‘(ℑ‘𝐴)))
2 cnrefiisplem.a . . . . . . . . 9 (𝜑𝐴 ∈ ℂ)
3 cnrefiisplem.n . . . . . . . . 9 (𝜑 → ¬ 𝐴 ∈ ℝ)
42, 3absimnre 43798 . . . . . . . 8 (𝜑 → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
54adantr 482 . . . . . . 7 ((𝜑𝑤 = (abs‘(ℑ‘𝐴))) → (abs‘(ℑ‘𝐴)) ∈ ℝ+)
61, 5eqeltrd 2834 . . . . . 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 2826 . . . . . . . . . . 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 4627 . . . . . . . . . 10 (𝑤 ≠ (abs‘(ℑ‘𝐴)) → ¬ 𝑤 ∈ {(abs‘(ℑ‘𝐴))})
13 elunnel1 4110 . . . . . . . . . 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 4959 . . . . . . . . 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 4603 . . . . . . . . 9 (𝑤 ∈ {(abs‘(𝑦𝐴))} ↔ 𝑤 = (abs‘(𝑦𝐴)))
1817rexbii 3094 . . . . . . . 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 486 . . . . . . . 8 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑤 = (abs‘(𝑦𝐴)))
22 eldifi 4087 . . . . . . . . . . . 12 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦 ∈ (𝐵 ∩ ℂ))
2322elin2d 4160 . . . . . . . . . . 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 11517 . . . . . . . . 9 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (𝑦𝐴) ∈ ℂ)
27 eldifsni 4751 . . . . . . . . . . 11 (𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) → 𝑦𝐴)
2827ad2antlr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑦𝐴)
2924, 25, 28subne0d 11526 . . . . . . . . 9 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (𝑦𝐴) ≠ 0)
3026, 29absrpcld 15339 . . . . . . . 8 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → (abs‘(𝑦𝐴)) ∈ ℝ+)
3121, 30eqeltrd 2834 . . . . . . 7 (((𝜑𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴})) ∧ 𝑤 = (abs‘(𝑦𝐴))) → 𝑤 ∈ ℝ+)
3231rexlimdva2 3151 . . . . . 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 3029 . . . 4 ((𝜑𝑤𝐷) → 𝑤 ∈ ℝ+)
3534ssd 43378 . . 3 (𝜑𝐷 ⊆ ℝ+)
36 cnrefiisplem.x . . . 4 𝑋 = inf(𝐷, ℝ*, < )
37 xrltso 13066 . . . . . 6 < Or ℝ*
3837a1i 11 . . . . 5 (𝜑 → < Or ℝ*)
39 snfi 8991 . . . . . . . 8 {(abs‘(ℑ‘𝐴))} ∈ Fin
4039a1i 11 . . . . . . 7 (𝜑 → {(abs‘(ℑ‘𝐴))} ∈ Fin)
41 cnrefiisplem.b . . . . . . . . 9 (𝜑𝐵 ∈ Fin)
42 inss1 4189 . . . . . . . . . . 11 (𝐵 ∩ ℂ) ⊆ 𝐵
4342a1i 11 . . . . . . . . . 10 (𝜑 → (𝐵 ∩ ℂ) ⊆ 𝐵)
4443ssdifssd 4103 . . . . . . . . 9 (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ⊆ 𝐵)
4541, 44ssfid 9214 . . . . . . . 8 (𝜑 → ((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin)
46 snfi 8991 . . . . . . . . 9 {(abs‘(𝑦𝐴))} ∈ Fin
4746rgenw 3065 . . . . . . . 8 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin
48 iunfi 9287 . . . . . . . 8 ((((𝐵 ∩ ℂ) ∖ {𝐴}) ∈ Fin ∧ ∀𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin)
4945, 47, 48sylancl 587 . . . . . . 7 (𝜑 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))} ∈ Fin)
5040, 49unfid 43451 . . . . . 6 (𝜑 → ({(abs‘(ℑ‘𝐴))} ∪ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}) ∈ Fin)
519, 50eqeltrid 2838 . . . . 5 (𝜑𝐷 ∈ Fin)
52 fvex 6856 . . . . . . . . . 10 (abs‘(ℑ‘𝐴)) ∈ V
5352snid 4623 . . . . . . . . 9 (abs‘(ℑ‘𝐴)) ∈ {(abs‘(ℑ‘𝐴))}
54 elun1 4137 . . . . . . . . 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 2833 . . . . . . 7 (abs‘(ℑ‘𝐴)) ∈ 𝐷
5756a1i 11 . . . . . 6 (𝜑 → (abs‘(ℑ‘𝐴)) ∈ 𝐷)
5857ne0d 4296 . . . . 5 (𝜑𝐷 ≠ ∅)
59 rpssxr 43802 . . . . . 6 + ⊆ ℝ*
6035, 59sstrdi 3957 . . . . 5 (𝜑𝐷 ⊆ ℝ*)
61 fiinfcl 9442 . . . . 5 (( < Or ℝ* ∧ (𝐷 ∈ Fin ∧ 𝐷 ≠ ∅ ∧ 𝐷 ⊆ ℝ*)) → inf(𝐷, ℝ*, < ) ∈ 𝐷)
6238, 51, 58, 60, 61syl13anc 1373 . . . 4 (𝜑 → inf(𝐷, ℝ*, < ) ∈ 𝐷)
6336, 62eqeltrid 2838 . . 3 (𝜑𝑋𝐷)
6435, 63sseldd 3946 . 2 (𝜑𝑋 ∈ ℝ+)
6535, 62sseldd 3946 . . . . . . . . . 10 (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ+)
6665rpred 12962 . . . . . . . . 9 (𝜑 → inf(𝐷, ℝ*, < ) ∈ ℝ)
6766adantr 482 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ∈ ℝ)
682imcld 15086 . . . . . . . . . . 11 (𝜑 → (ℑ‘𝐴) ∈ ℝ)
6968recnd 11188 . . . . . . . . . 10 (𝜑 → (ℑ‘𝐴) ∈ ℂ)
7069adantr 482 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (ℑ‘𝐴) ∈ ℂ)
7170abscld 15327 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ∈ ℝ)
72 recn 11146 . . . . . . . . . . 11 (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
7372adantl 483 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
742adantr 482 . . . . . . . . . 10 ((𝜑𝑦 ∈ ℝ) → 𝐴 ∈ ℂ)
7573, 74subcld 11517 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → (𝑦𝐴) ∈ ℂ)
7675abscld 15327 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(𝑦𝐴)) ∈ ℝ)
7760adantr 482 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 𝐷 ⊆ ℝ*)
78 infxrlb 13259 . . . . . . . . 9 ((𝐷 ⊆ ℝ* ∧ (abs‘(ℑ‘𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
7977, 56, 78sylancl 587 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(ℑ‘𝐴)))
80 simpr 486 . . . . . . . . 9 ((𝜑𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
8174, 80absimlere 43801 . . . . . . . 8 ((𝜑𝑦 ∈ ℝ) → (abs‘(ℑ‘𝐴)) ≤ (abs‘(𝑦𝐴)))
8267, 71, 76, 79, 81letrd 11317 . . . . . . 7 ((𝜑𝑦 ∈ ℝ) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
8336, 82eqbrtrid 5141 . . . . . 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 2826 . . . . . . . 8 (𝑦𝐶𝑦 ∈ (ℝ ∪ 𝐵))
87 elunnel1 4110 . . . . . . . 8 ((𝑦 ∈ (ℝ ∪ 𝐵) ∧ ¬ 𝑦 ∈ ℝ) → 𝑦𝐵)
8886, 87sylanb 582 . . . . . . 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 486 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦𝐵)
92 simpll 766 . . . . . . . . . . . . . . . 16 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ ℂ)
9391, 92elind 4155 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ (𝐵 ∩ ℂ))
94 nelsn 4627 . . . . . . . . . . . . . . . 16 (𝑦𝐴 → ¬ 𝑦 ∈ {𝐴})
9594ad2antlr 726 . . . . . . . . . . . . . . 15 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → ¬ 𝑦 ∈ {𝐴})
9693, 95eldifd 3922 . . . . . . . . . . . . . 14 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}))
97 fvex 6856 . . . . . . . . . . . . . . 15 (abs‘(𝑦𝐴)) ∈ V
9897snid 4623 . . . . . . . . . . . . . 14 (abs‘(𝑦𝐴)) ∈ {(abs‘(𝑦𝐴))}
99 fvoveq1 7381 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑦 → (abs‘(𝑤𝐴)) = (abs‘(𝑦𝐴)))
10099sneqd 4599 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → {(abs‘(𝑤𝐴))} = {(abs‘(𝑦𝐴))})
101100eliuni 4961 . . . . . . . . . . . . . 14 ((𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}) ∧ (abs‘(𝑦𝐴)) ∈ {(abs‘(𝑦𝐴))}) → (abs‘(𝑦𝐴)) ∈ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))})
10296, 98, 101sylancl 587 . . . . . . . . . . . . 13 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))})
103100cbviunv 5001 . . . . . . . . . . . . 13 𝑤 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑤𝐴))} = 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))}
104102, 103eleqtrdi 2844 . . . . . . . . . . . 12 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝑦 ∈ ((𝐵 ∩ ℂ) ∖ {𝐴}){(abs‘(𝑦𝐴))})
105 elun2 4138 . . . . . . . . . . . 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 2845 . . . . . . . . . 10 (((𝑦 ∈ ℂ ∧ 𝑦𝐴) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝐷)
108107adantll 713 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → (abs‘(𝑦𝐴)) ∈ 𝐷)
109 infxrlb 13259 . . . . . . . . 9 ((𝐷 ⊆ ℝ* ∧ (abs‘(𝑦𝐴)) ∈ 𝐷) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
11090, 108, 109syl2anc 585 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → inf(𝐷, ℝ*, < ) ≤ (abs‘(𝑦𝐴)))
11136, 110eqbrtrid 5141 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝑋 ≤ (abs‘(𝑦𝐴)))
112111adantllr 718 . . . . . 6 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ 𝑦𝐵) → 𝑋 ≤ (abs‘(𝑦𝐴)))
11389, 112syldan 592 . . . . 5 ((((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) ∧ ¬ 𝑦 ∈ ℝ) → 𝑋 ≤ (abs‘(𝑦𝐴)))
11484, 113pm2.61dan 812 . . . 4 (((𝜑𝑦𝐶) ∧ (𝑦 ∈ ℂ ∧ 𝑦𝐴)) → 𝑋 ≤ (abs‘(𝑦𝐴)))
115114ex 414 . . 3 ((𝜑𝑦𝐶) → ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴))))
116115ralrimiva 3140 . 2 (𝜑 → ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴))))
117 breq1 5109 . . . . 5 (𝑥 = 𝑋 → (𝑥 ≤ (abs‘(𝑦𝐴)) ↔ 𝑋 ≤ (abs‘(𝑦𝐴))))
118117imbi2d 341 . . . 4 (𝑥 = 𝑋 → (((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))) ↔ ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))))
119118ralbidv 3171 . . 3 (𝑥 = 𝑋 → (∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))) ↔ ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))))
120119rspcev 3580 . 2 ((𝑋 ∈ ℝ+ ∧ ∀𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑋 ≤ (abs‘(𝑦𝐴)))) → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
12164, 116, 120syl2anc 585 1 (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝐶 ((𝑦 ∈ ℂ ∧ 𝑦𝐴) → 𝑥 ≤ (abs‘(𝑦𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  cdif 3908  cun 3909  cin 3910  wss 3911  c0 4283  {csn 4587   ciun 4955   class class class wbr 5106   Or wor 5545  cfv 6497  (class class class)co 7358  Fincfn 8886  infcinf 9382  cc 11054  cr 11055  *cxr 11193   < clt 11194  cle 11195  cmin 11390  +crp 12920  cim 14989  abscabs 15125
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-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  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-seq 13913  df-exp 13974  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127
This theorem is referenced by:  cnrefiisp  44157
  Copyright terms: Public domain W3C validator