Theorem ulm2 26363

Description: Simplify ulmval 26358 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 26358	. . 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 1096	. . . 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 6672	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐹 = 𝑍)
8		fdm 6671	. . . . . . . . . . 11 ⊢ (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) → dom 𝐹 = (ℤ≥‘𝑛))
9	7, 8	sylan9req 2793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑍 = (ℤ≥‘𝑛))
10	5, 9	eqtr3id 2786	. . . . . . . . 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 12804	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑀 = 𝑛)
16	15	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆)) → 𝑛 = 𝑀)
17		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (ℤ≥‘𝑛) = (ℤ≥‘𝑀))
18	17, 5	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (ℤ≥‘𝑛) = 𝑍)
19	18	feq2d 6646	. . . . . . . . 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 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆))
22	16, 21	impbida 801	. . . . . 6 ⊢ (𝜑 → (𝐹:(ℤ≥‘𝑛)⟶(ℂ ↑m 𝑆) ↔ 𝑛 = 𝑀))
23	22	anbi1d 632	. . . . 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 783	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → 𝜑)
27		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑛 = 𝑀)
28		uzid 12794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
29	11, 28	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
30	29, 5	eleqtrrdi 2848	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑀 ∈ 𝑍)
31	30	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑀 ∈ 𝑍)
32	27, 31	eqeltrd 2837	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑛 = 𝑀) → 𝑛 ∈ 𝑍)
33	5	uztrn2 12798	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
34	32, 33	sylan 581	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	5	uztrn2 12798	. . . . . . . . . . . . . . . 16 ⊢ ((𝑗 ∈ 𝑍 ∧ 𝑘 ∈ (ℤ≥‘𝑗)) → 𝑘 ∈ 𝑍)
36	34, 35	sylan 581	. . . . . . . . . . . . . . 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 837	. . . . . . . . . . . . 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 589	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (𝐺‘𝑧) = 𝐴)
43	40, 42	oveq12d 7378	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧)) = (𝐵 − 𝐴))
44	43	fveq2d 6838	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑛 = 𝑀) ∧ 𝑗 ∈ (ℤ≥‘𝑛)) ∧ 𝑘 ∈ (ℤ≥‘𝑗)) ∧ 𝑧 ∈ 𝑆) → (abs‘(((𝐹‘𝑘)‘𝑧) − (𝐺‘𝑧))) = (abs‘(𝐵 − 𝐴)))
45	44	breq1d 5096	. . . . . . . . . 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 3159	. . . . . . . . 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 3159	. . . . . . . 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 3160	. . . . . . 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 3161	. . . . . 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 579	. . . . 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 309	. . . 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	bitrid 283	. . 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 3162	. 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 3297	. . . . 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 3161	. . . 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 3598	. . 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 305	1 ⊢ (𝜑 → (𝐹(⇝𝑢‘𝑆)𝐺 ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑧 ∈ 𝑆 (abs‘(𝐵 − 𝐴)) < 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   class class class wbr 5086  dom cdm 5624  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ↑_m cmap 8766  ℂcc 11027   < clt 11170   − cmin 11368  ℤcz 12515  ℤ_≥cuz 12779  ℝ⁺crp 12933  abscabs 15187  ⇝_𝑢culm 26354

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-er 8636  df-map 8768  df-pm 8769  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-neg 11371  df-z 12516  df-uz 12780  df-ulm 26355

This theorem is referenced by:  ulmi  26364  ulmclm  26365  ulmres  26366  ulmshftlem  26367  ulm0  26369  ulmcau  26373  ulmss  26375