Theorem ulm2 25449

Description: Simplify ulmval 25444 when 𝐹 and 𝐺 are known to be functions. (Contributed by Mario Carneiro, 26-Feb-2015.)

Hypotheses

Ref	Expression
ulm2.z	⊢ 𝑍 = (ℤ≥‘𝑀)
ulm2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
ulm2.f	⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
ulm2.b	⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
ulm2.a	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
ulm2.g	⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
ulm2.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)

Assertion

Ref	Expression
ulm2	⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 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.

Step	Hyp	Ref	Expression
1		ulm2.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
2		ulmval 25444	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
4		3anan12 1094	. . . 4 ⊢ ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
5		ulm2.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		ulm2.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(ℂ ↑m 𝑆))
7	6	fdmd 6595	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝑍)
8		fdm 6593	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) → dom 𝐹 = (ℤ≥‘𝑛))
9	7, 8	sylan9req 2800	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑍 = (ℤ≥‘𝑛))
10	5, 9	eqtr3id 2793	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → (ℤ≥‘𝑀) = (ℤ≥‘𝑛))
11		ulm2.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
12	11	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 ∈ ℤ)
13		uz11 12536	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝑛) ↔ 𝑀 = 𝑛))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → ((ℤ≥‘𝑀) = (ℤ≥‘𝑛) ↔ 𝑀 = 𝑛))
15	10, 14	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 = 𝑛)
16	15	eqcomd 2744	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑛 = 𝑀)
17		fveq2 6756	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (ℤ≥‘𝑛) = (ℤ≥‘𝑀))
18	17, 5	eqtr4di 2797	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (ℤ≥‘𝑛) = 𝑍)
19	18	feq2d 6570	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝐹:𝑍⟶(ℂ ↑m 𝑆)))
20	19	biimparc 479	. . . . . . . 8 ⊢ ((𝐹:𝑍⟶(ℂ ↑m 𝑆) ∧ 𝑛 = 𝑀) → 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆))
21	6, 20	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆))
22	16, 21	impbida 797	. . . . . 6 ⊢ (𝜑 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝑛 = 𝑀))
23	22	anbi1d 629	. . . . 5 ⊢ (𝜑 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)))
24		ulm2.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑆⟶ℂ)
25	24	biantrurd 532	. . . . 5 ⊢ (𝜑 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥))))
26		simp-4l 779	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝜑)
27		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
28		uzid 12526	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
29	11, 28	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
30	29, 5	eleqtrrdi 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
31	30	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑀 ∈ 𝑍)
32	27, 31	eqeltrd 2839	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑛 ∈ 𝑍)
33	5	uztrn2 12530	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
34	32, 33	sylan 579	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	5	uztrn2 12530	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
36	34, 35	sylan 579	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝑘 ∈ 𝑍)
38		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝑧 ∈ 𝑆)
39		ulm2.b	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝑍 ∧ 𝑧 ∈ 𝑆)) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
40	26, 37, 38, 39	syl12anc 833	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((𝐹‘𝑘)‘𝑧) = 𝐵)
41		ulm2.a	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
42	26, 41	sylancom 587	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
43	40, 42	oveq12d 7273	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (𝐵 − 𝐴))
44	43	fveq2d 6760	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(𝐵 − 𝐴)))
45	44	breq1d 5080	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → ((abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ (abs‘(𝐵 − 𝐴)) < 𝑥))
46	45	ralbidva 3119	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → (∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
47	46	ralbidva 3119	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → (∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
48	47	rexbidva 3224	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
49	48	ralbidv 3120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
50	49	pm5.32da 578	. . . . 5 ⊢ (𝜑 → ((𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
51	23, 25, 50	3bitr3d 308	. . . 4 ⊢ (𝜑 → ((𝐺:𝑆⟶ℂ ∧ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥)) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
52	4, 51	syl5bb 282	. . 3 ⊢ (𝜑 → ((𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
53	52	rexbidv 3225	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℤ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ∧ 𝐺:𝑆⟶ℂ ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) < 𝑥) ↔ ∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥)))
54	18	rexeqdv 3340	. . . . 5 ⊢ (𝑛 = 𝑀 → (∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
55	54	ralbidv 3120	. . . 4 ⊢ (𝑛 = 𝑀 → (∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
56	55	ceqsrexv 3578	. . 3 ⊢ (𝑀 ∈ ℤ → (∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
57	11, 56	syl 17	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℤ (𝑛 = 𝑀 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ (ℤ≥‘𝑛)∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))
58	3, 53, 57	3bitrd 304	1 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   class class class wbr 5070  dom cdm 5580  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  ℂcc 10800   < clt 10940   − cmin 11135  ℤcz 12249  ℤ_≥cuz 12511  ℝ⁺crp 12659  abscabs 14873  ⇝_𝑢culm 25440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-pre-lttri 10876  ax-pre-lttrn 10877

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-er 8456  df-map 8575  df-pm 8576  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-neg 11138  df-z 12250  df-uz 12512  df-ulm 25441

This theorem is referenced by:  ulmi  25450  ulmclm  25451  ulmres  25452  ulmshftlem  25453  ulm0  25455  ulmcau  25459  ulmss  25461