MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ulm2 Structured version   Visualization version   GIF version

Theorem ulm2 25744
Description: Simplify ulmval 25739 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)
Hypotheses
Ref Expression
ulm2.z 𝑍 = (ℤ𝑀)
ulm2.m (𝜑𝑀 ∈ ℤ)
ulm2.f (𝜑𝐹:𝑍⟶(ℂ ↑m 𝑆))
ulm2.b ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)
ulm2.a ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)
ulm2.g (𝜑𝐺:𝑆⟶ℂ)
ulm2.s (𝜑𝑆𝑉)
Assertion
Ref Expression
ulm2 (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
Distinct variable groups:   𝑗,𝑘,𝑥,𝑧,𝐹   𝑗,𝐺,𝑘,𝑥,𝑧   𝑗,𝑀,𝑘,𝑥,𝑧   𝜑,𝑗,𝑘,𝑥,𝑧   𝐴,𝑗,𝑘,𝑥   𝑥,𝐵   𝑆,𝑗,𝑘,𝑥,𝑧   𝑗,𝑍,𝑥
Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑧,𝑗,𝑘)   𝑉(𝑥,𝑧,𝑗,𝑘)   𝑍(𝑧,𝑘)

Proof of Theorem ulm2
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 ulm2.s . . 3 (𝜑𝑆𝑉)
2 ulmval 25739 . . 3 (𝑆𝑉 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
31, 2syl 17 . 2 (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
4 3anan12 1096 . . . 4 ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) ↔ (𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
5 ulm2.z . . . . . . . . . 10 𝑍 = (ℤ𝑀)
6 ulm2.f . . . . . . . . . . . 12 (𝜑𝐹:𝑍⟶(ℂ ↑m 𝑆))
76fdmd 6679 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝑍)
8 fdm 6677 . . . . . . . . . . 11 (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) → dom 𝐹 = (ℤ𝑛))
97, 8sylan9req 2797 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)) → 𝑍 = (ℤ𝑛))
105, 9eqtr3id 2790 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)) → (ℤ𝑀) = (ℤ𝑛))
11 ulm2.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1211adantr 481 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 ∈ ℤ)
13 uz11 12788 . . . . . . . . . 10 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑛) ↔ 𝑀 = 𝑛))
1412, 13syl 17 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)) → ((ℤ𝑀) = (ℤ𝑛) ↔ 𝑀 = 𝑛))
1510, 14mpbid 231 . . . . . . . 8 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 = 𝑛)
1615eqcomd 2742 . . . . . . 7 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆)) → 𝑛 = 𝑀)
17 fveq2 6842 . . . . . . . . . . 11 (𝑛 = 𝑀 → (ℤ𝑛) = (ℤ𝑀))
1817, 5eqtr4di 2794 . . . . . . . . . 10 (𝑛 = 𝑀 → (ℤ𝑛) = 𝑍)
1918feq2d 6654 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑m 𝑆)))
2019biimparc 480 . . . . . . . 8 ((𝐹:𝑍⟶(ℂ ↑m 𝑆) ∧ 𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆))
216, 20sylan 580 . . . . . . 7 ((𝜑𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆))
2216, 21impbida 799 . . . . . 6 (𝜑 → (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝑛 = 𝑀))
2322anbi1d 630 . . . . 5 (𝜑 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)))
24 ulm2.g . . . . . 6 (𝜑𝐺:𝑆⟶ℂ)
2524biantrurd 533 . . . . 5 (𝜑 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) ↔ (𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥))))
26 simp-4l 781 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝜑)
27 simpr 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑛 = 𝑀)
28 uzid 12778 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
2911, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ𝑀))
3029, 5eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀𝑍)
3130adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑀𝑍)
3227, 31eqeltrd 2838 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 = 𝑀) → 𝑛𝑍)
335uztrn2 12782 . . . . . . . . . . . . . . . . 17 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
3432, 33sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
355uztrn2 12782 . . . . . . . . . . . . . . . 16 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3634, 35sylan 580 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3736adantr 481 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑘𝑍)
38 simpr 485 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑧𝑆)
39 ulm2.b . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)
4026, 37, 38, 39syl12anc 835 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((𝐹𝑘)‘𝑧) = 𝐵)
41 ulm2.a . . . . . . . . . . . . . 14 ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)
4226, 41sylancom 588 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (𝐺𝑧) = 𝐴)
4340, 42oveq12d 7375 . . . . . . . . . . . 12 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (((𝐹𝑘)‘𝑧) − (𝐺𝑧)) = (𝐵𝐴))
4443fveq2d 6846 . . . . . . . . . . 11 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) = (abs‘(𝐵𝐴)))
4544breq1d 5115 . . . . . . . . . 10 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
4645ralbidva 3172 . . . . . . . . 9 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → (∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4746ralbidva 3172 . . . . . . . 8 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → (∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4847rexbidva 3173 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4948ralbidv 3174 . . . . . 6 ((𝜑𝑛 = 𝑀) → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5049pm5.32da 579 . . . . 5 (𝜑 → ((𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥)))
5123, 25, 503bitr3d 308 . . . 4 (𝜑 → ((𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥)) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥)))
524, 51bitrid 282 . . 3 (𝜑 → ((𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥)))
5352rexbidv 3175 . 2 (𝜑 → (∃𝑛 ∈ ℤ (𝐹:(ℤ𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥)))
5418rexeqdv 3314 . . . . 5 (𝑛 = 𝑀 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5554ralbidv 3174 . . . 4 (𝑛 = 𝑀 → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5655ceqsrexv 3605 . . 3 (𝑀 ∈ ℤ → (∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5711, 56syl 17 . 2 (𝜑 → (∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
583, 53, 573bitrd 304 1 (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073   class class class wbr 5105  dom cdm 5633  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cc 11049   < clt 11189  cmin 11385  cz 12499  cuz 12763  +crp 12915  abscabs 15119  𝑢culm 25735
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-neg 11388  df-z 12500  df-uz 12764  df-ulm 25736
This theorem is referenced by:  ulmi  25745  ulmclm  25746  ulmres  25747  ulmshftlem  25748  ulm0  25750  ulmcau  25754  ulmss  25756
  Copyright terms: Public domain W3C validator