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Theorem ulm2 24043
Description: Simplify ulmval 24038 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 24038 . . 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 1049 . . . 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))
7 fdm 6008 . . . . . . . . . . . 12 (𝐹:𝑍⟶(ℂ ↑𝑚 𝑆) → dom 𝐹 = 𝑍)
86, 7syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝑍)
9 fdm 6008 . . . . . . . . . . 11 (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → dom 𝐹 = (ℤ𝑛))
108, 9sylan9req 2676 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑍 = (ℤ𝑛))
115, 10syl5eqr 2669 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → (ℤ𝑀) = (ℤ𝑛))
12 ulm2.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1312adantr 481 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑀 ∈ ℤ)
14 uz11 11654 . . . . . . . . . 10 (𝑀 ∈ ℤ → ((ℤ𝑀) = (ℤ𝑛) ↔ 𝑀 = 𝑛))
1513, 14syl 17 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → ((ℤ𝑀) = (ℤ𝑛) ↔ 𝑀 = 𝑛))
1611, 15mpbid 222 . . . . . . . 8 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑀 = 𝑛)
1716eqcomd 2627 . . . . . . 7 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑛 = 𝑀)
18 fveq2 6148 . . . . . . . . . . 11 (𝑛 = 𝑀 → (ℤ𝑛) = (ℤ𝑀))
1918, 5syl6eqr 2673 . . . . . . . . . 10 (𝑛 = 𝑀 → (ℤ𝑛) = 𝑍)
2019feq2d 5988 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)))
2120biimparc 504 . . . . . . . 8 ((𝐹:𝑍⟶(ℂ ↑𝑚 𝑆) ∧ 𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
226, 21sylan 488 . . . . . . 7 ((𝜑𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
2317, 22impbida 876 . . . . . 6 (𝜑 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝑛 = 𝑀))
2423anbi1d 740 . . . . 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)))
25 ulm2.g . . . . . 6 (𝜑𝐺:𝑆⟶ℂ)
2625biantrurd 529 . . . . 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))))
27 simp-4l 805 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝜑)
28 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑛 = 𝑀)
29 uzid 11646 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3012, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ𝑀))
3130, 5syl6eleqr 2709 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀𝑍)
3231adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑀𝑍)
3328, 32eqeltrd 2698 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 = 𝑀) → 𝑛𝑍)
345uztrn2 11649 . . . . . . . . . . . . . . . . 17 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
3533, 34sylan 488 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
365uztrn2 11649 . . . . . . . . . . . . . . . 16 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3735, 36sylan 488 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3837adantr 481 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑘𝑍)
39 simpr 477 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑧𝑆)
40 ulm2.b . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)
4127, 38, 39, 40syl12anc 1321 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((𝐹𝑘)‘𝑧) = 𝐵)
42 ulm2.a . . . . . . . . . . . . . 14 ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)
4327, 42sylancom 700 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (𝐺𝑧) = 𝐴)
4441, 43oveq12d 6622 . . . . . . . . . . . 12 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (((𝐹𝑘)‘𝑧) − (𝐺𝑧)) = (𝐵𝐴))
4544fveq2d 6152 . . . . . . . . . . 11 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) = (abs‘(𝐵𝐴)))
4645breq1d 4623 . . . . . . . . . 10 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
4746ralbidva 2979 . . . . . . . . 9 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → (∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4847ralbidva 2979 . . . . . . . 8 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → (∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4948rexbidva 3042 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5049ralbidv 2980 . . . . . 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))
5150pm5.32da 672 . . . . 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)))
5224, 26, 513bitr3d 298 . . . 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)))
534, 52syl5bb 272 . . 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)))
5453rexbidv 3045 . 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)))
5519rexeqdv 3134 . . . . 5 (𝑛 = 𝑀 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5655ralbidv 2980 . . . 4 (𝑛 = 𝑀 → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5756ceqsrexv 3319 . . 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))
5812, 57syl 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))
593, 54, 583bitrd 294 1 (𝜑 → (𝐹(⇝𝑢𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908   class class class wbr 4613  dom cdm 5074  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802  cc 9878   < clt 10018  cmin 10210  cz 11321  cuz 11631  +crp 11776  abscabs 13908  𝑢culm 24034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-neg 10213  df-z 11322  df-uz 11632  df-ulm 24035
This theorem is referenced by:  ulmi  24044  ulmclm  24045  ulmres  24046  ulmshftlem  24047  ulm0  24049  ulmcau  24053  ulmss  24055
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