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Theorem ulm2 24430
Description: Simplify ulmval 24425 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 24425 . . 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 1117 . . . 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 6232 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝑍)
8 fdm 6231 . . . . . . . . . . 11 (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → dom 𝐹 = (ℤ𝑛))
97, 8sylan9req 2820 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑍 = (ℤ𝑛))
105, 9syl5eqr 2813 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → (ℤ𝑀) = (ℤ𝑛))
11 ulm2.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1211adantr 472 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑀 ∈ ℤ)
13 uz11 11909 . . . . . . . . . 10 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑛) ↔ 𝑀 = 𝑛))
1412, 13syl 17 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → ((ℤ𝑀) = (ℤ𝑛) ↔ 𝑀 = 𝑛))
1510, 14mpbid 223 . . . . . . . 8 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑀 = 𝑛)
1615eqcomd 2771 . . . . . . 7 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑛 = 𝑀)
17 fveq2 6375 . . . . . . . . . . 11 (𝑛 = 𝑀 → (ℤ𝑛) = (ℤ𝑀))
1817, 5syl6eqr 2817 . . . . . . . . . 10 (𝑛 = 𝑀 → (ℤ𝑛) = 𝑍)
1918feq2d 6209 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)))
2019biimparc 471 . . . . . . . 8 ((𝐹:𝑍⟶(ℂ ↑𝑚 𝑆) ∧ 𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
216, 20sylan 575 . . . . . . 7 ((𝜑𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
2216, 21impbida 835 . . . . . 6 (𝜑 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝑛 = 𝑀))
2322anbi1d 623 . . . . 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 528 . . . . 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 801 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝜑)
27 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑛 = 𝑀)
28 uzid 11901 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
2911, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ𝑀))
3029, 5syl6eleqr 2855 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀𝑍)
3130adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑀𝑍)
3227, 31eqeltrd 2844 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 = 𝑀) → 𝑛𝑍)
335uztrn2 11904 . . . . . . . . . . . . . . . . 17 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
3432, 33sylan 575 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
355uztrn2 11904 . . . . . . . . . . . . . . . 16 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3634, 35sylan 575 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3736adantr 472 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑘𝑍)
38 simpr 477 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑧𝑆)
39 ulm2.b . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)
4026, 37, 38, 39syl12anc 865 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((𝐹𝑘)‘𝑧) = 𝐵)
41 ulm2.a . . . . . . . . . . . . . 14 ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)
4226, 41sylancom 582 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (𝐺𝑧) = 𝐴)
4340, 42oveq12d 6860 . . . . . . . . . . . 12 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (((𝐹𝑘)‘𝑧) − (𝐺𝑧)) = (𝐵𝐴))
4443fveq2d 6379 . . . . . . . . . . 11 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) = (abs‘(𝐵𝐴)))
4544breq1d 4819 . . . . . . . . . 10 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
4645ralbidva 3132 . . . . . . . . 9 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → (∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4746ralbidva 3132 . . . . . . . 8 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → (∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4847rexbidva 3196 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4948ralbidv 3133 . . . . . 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 574 . . . . 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 300 . . . 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, 51syl5bb 274 . . 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 3199 . 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 3293 . . . . 5 (𝑛 = 𝑀 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5554ralbidv 3133 . . . 4 (𝑛 = 𝑀 → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5655ceqsrexv 3489 . . 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 296 1 (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056   class class class wbr 4809  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cc 10187   < clt 10328  cmin 10520  cz 11624  cuz 11886  +crp 12028  abscabs 14261  𝑢culm 24421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-pre-lttri 10263  ax-pre-lttrn 10264
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-po 5198  df-so 5199  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-neg 10523  df-z 11625  df-uz 11887  df-ulm 24422
This theorem is referenced by:  ulmi  24431  ulmclm  24432  ulmres  24433  ulmshftlem  24434  ulm0  24436  ulmcau  24440  ulmss  24442
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