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Theorem ulm2 24338
 Description: Simplify ulmval 24333 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 24333 . . 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 1082 . . . 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 6212 . . . . . . . . . . . 12 (𝐹:𝑍⟶(ℂ ↑𝑚 𝑆) → dom 𝐹 = 𝑍)
86, 7syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝑍)
9 fdm 6212 . . . . . . . . . . 11 (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) → dom 𝐹 = (ℤ𝑛))
108, 9sylan9req 2815 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑍 = (ℤ𝑛))
115, 10syl5eqr 2808 . . . . . . . . 9 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → (ℤ𝑀) = (ℤ𝑛))
12 ulm2.m . . . . . . . . . . 11 (𝜑𝑀 ∈ ℤ)
1312adantr 472 . . . . . . . . . 10 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑀 ∈ ℤ)
14 uz11 11902 . . . . . . . . . 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 2766 . . . . . . 7 ((𝜑𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆)) → 𝑛 = 𝑀)
18 fveq2 6352 . . . . . . . . . . 11 (𝑛 = 𝑀 → (ℤ𝑛) = (ℤ𝑀))
1918, 5syl6eqr 2812 . . . . . . . . . 10 (𝑛 = 𝑀 → (ℤ𝑛) = 𝑍)
2019feq2d 6192 . . . . . . . . 9 (𝑛 = 𝑀 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑𝑚 𝑆)))
2120biimparc 505 . . . . . . . 8 ((𝐹:𝑍⟶(ℂ ↑𝑚 𝑆) ∧ 𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
226, 21sylan 489 . . . . . . 7 ((𝜑𝑛 = 𝑀) → 𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆))
2317, 22impbida 913 . . . . . 6 (𝜑 → (𝐹:(ℤ𝑛)⟶(ℂ ↑𝑚 𝑆) ↔ 𝑛 = 𝑀))
2423anbi1d 743 . . . . 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 530 . . . . 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 825 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝜑)
28 simpr 479 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑛 = 𝑀)
29 uzid 11894 . . . . . . . . . . . . . . . . . . . . 21 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
3012, 29syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑀 ∈ (ℤ𝑀))
3130, 5syl6eleqr 2850 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑀𝑍)
3231adantr 472 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑛 = 𝑀) → 𝑀𝑍)
3328, 32eqeltrd 2839 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 = 𝑀) → 𝑛𝑍)
345uztrn2 11897 . . . . . . . . . . . . . . . . 17 ((𝑛𝑍𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
3533, 34sylan 489 . . . . . . . . . . . . . . . 16 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → 𝑗𝑍)
365uztrn2 11897 . . . . . . . . . . . . . . . 16 ((𝑗𝑍𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3735, 36sylan 489 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → 𝑘𝑍)
3837adantr 472 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑘𝑍)
39 simpr 479 . . . . . . . . . . . . . 14 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → 𝑧𝑆)
40 ulm2.b . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑘𝑍𝑧𝑆)) → ((𝐹𝑘)‘𝑧) = 𝐵)
4127, 38, 39, 40syl12anc 1475 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((𝐹𝑘)‘𝑧) = 𝐵)
42 ulm2.a . . . . . . . . . . . . . 14 ((𝜑𝑧𝑆) → (𝐺𝑧) = 𝐴)
4327, 42sylancom 704 . . . . . . . . . . . . 13 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (𝐺𝑧) = 𝐴)
4441, 43oveq12d 6831 . . . . . . . . . . . 12 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (((𝐹𝑘)‘𝑧) − (𝐺𝑧)) = (𝐵𝐴))
4544fveq2d 6356 . . . . . . . . . . 11 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) = (abs‘(𝐵𝐴)))
4645breq1d 4814 . . . . . . . . . 10 (((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) ∧ 𝑧𝑆) → ((abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ (abs‘(𝐵𝐴)) < 𝑥))
4746ralbidva 3123 . . . . . . . . 9 ((((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) ∧ 𝑘 ∈ (ℤ𝑗)) → (∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4847ralbidva 3123 . . . . . . . 8 (((𝜑𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ𝑛)) → (∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
4948rexbidva 3187 . . . . . . 7 ((𝜑𝑛 = 𝑀) → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(((𝐹𝑘)‘𝑧) − (𝐺𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5049ralbidv 3124 . . . . . 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 676 . . . . 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 3190 . 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 3284 . . . . 5 (𝑛 = 𝑀 → (∃𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5655ralbidv 3124 . . . 4 (𝑛 = 𝑀 → (∀𝑥 ∈ ℝ+𝑗 ∈ (ℤ𝑛)∀𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)∀𝑧𝑆 (abs‘(𝐵𝐴)) < 𝑥))
5756ceqsrexv 3475 . . 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 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051   class class class wbr 4804  dom cdm 5266  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↑𝑚 cmap 8023  ℂcc 10126   < clt 10266   − cmin 10458  ℤcz 11569  ℤ≥cuz 11879  ℝ+crp 12025  abscabs 14173  ⇝𝑢culm 24329 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-pre-lttri 10202  ax-pre-lttrn 10203 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-po 5187  df-so 5188  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-neg 10461  df-z 11570  df-uz 11880  df-ulm 24330 This theorem is referenced by:  ulmi  24339  ulmclm  24340  ulmres  24341  ulmshftlem  24342  ulm0  24344  ulmcau  24348  ulmss  24350
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